Second order families of special Lagrangian submanifolds ...marianty/Papers/slagrC4_jdg.pdf ·...

j. differential geometry

65 (2003) 211-272

SECOND ORDER FAMILIES OF SPECIALLAGRANGIAN SUBMANIFOLDS IN C4

MARIANTY IONEL

AbstractThis paper extends to dimension 4 the results in the article “Second orderfamilies of special Lagrangian 3-folds” by Robert Bryant. We consider theproblem of classifying the special Lagrangian 4-folds in C4 whose funda-mental cubic at each point has a nontrivial stabilizer in SO(4). Points onspecial Lagrangian 4-folds where the SO(4)-stabilizer is nontrivial are theanalogs of the umbilical points in the classical theory of surfaces. In provingexistence for the families of special Lagrangian 4-folds, we used the methodof exterior differential systems in Cartan-Kahler theory. This method isguaranteed to tell us whether there are any families of special Lagrangiansubmanifolds with a certain stabilizer type, but does not give us an explicitdescription of the submanifolds. To derive an explicit description, we lookedat foliations by submanifolds and at other geometric particularities. In thismanner, we settled many of the cases and described the families of specialLagrangian submanifolds in an explicit way.

1. Introduction

The complex space Cm is endowed with a Kahler form

ω =i

2(dz1 ∧ dz1 + dz2 ∧ dz2 + · · · + dzm ∧ dzm)

and a volume form

Ω = dz1 ∧ dz2 ∧ · · · ∧ dzm,

where (z1, z2, . . . , zm) are the coordinates on Cm. A special Lagrangiansubmanifold in Cm is an m-dimensional real submanifold on which theforms ω and Im Ω restrict to 0.

Received 02/19/2003.

211

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The study of special Lagrangian (SL) submanifolds started with Har-vey and Lawson in their paper [12] on calibrated geometries. They con-structed many interesting examples of SL m-folds in Cm and provedlocal existence theorems. Since then, many other examples have beenconstructed using a variety of techniques. To give some examples, Do-minic Joyce used the method of ruled submanifolds, integrable systemsand evolution of quadrics in [8], [9], [10] to construct explicit examplesof special Lagrangian m-folds in Cm, Mark Haskins exhibited examplesof special Lagrangian cones in C3 [4], Richard Schoen and Jon Wolfsonused the variational approach for some of their constructions in Calabi-Yau manifolds in [15], etc.

Special Lagrangian geometry received reinforced attention in 1996when Strominger, Yau and Zaslow formulated what is today known asthe SYZ conjecture [17]. This conjecture reveals the role of the specialLagrangian geometry in mirror symmetry, a mysterious relationship be-tween pairs of Calabi-Yau 3-folds, coming from string theory. In thislarger context, a lot of research is going on nowadays to find examplesof special Lagrangian submanifolds. This would help in understandingwhat kind of singularities a special Lagrangian submanifold in a Calabi-Yau can have, classifying them and maybe ultimately resolving the SYZconjecture.

While, from the string theory point of view, the most interestingcase to study is the special Lagrangian 3-folds of a Calabi-Yau 3-fold,higher dimensional cases are also important for the understanding ofthe general theory of SL submanifolds in Calabi-Yau m-folds.

The idea in this research, initiated by Robert Bryant in his paper[1], is to classify families of SL submanifolds that are characterized byinvariant, geometric conditions. When the ambient space is flat, thesecond fundamental form is the lowest order invariant of a SL submani-fold, so we would like to study the second order families of SL m-folds inCm, that is the families of SL m-folds in Cm whose second fundamentalform satisfies a set of pointwise conditions.

The second fundamental form of a special Lagrangian submanifoldin Cm has a natural interpretation as a traceless cubic form on thesubmanifold, called the fundamental cubic. The stabilizer at a genericpoint of the fundamental cubic of a generic SL m-fold is trivial. Forcomparison, in the case of a hypersurface in Rm+1, the stabilizer of thesecond fundamental form in SO(m) is always nontrivial and is largerthan the minimum possible stabilizer exactly at the umbilical points ofthe hypersurface. For this reason, the points on SL m-folds where the

special lagrangian submanifolds 213

SO(m)-stabilizer is nontrivial are the analogs of the umbilical points inthe classical theory of surfaces.

In his article [1], Robert Bryant considered the ‘umbilical’ case andcompletely classified the SL submanifolds of C3 whose fundamental cu-bic has nontrivial SO(3)-stabilizer at a generic point. He found that theonly SL 3-folds whose fundamental cubic has a nontrivial stabilizer ata generic point are the 3-planes, with stabilizer SO(3), the Harvey andLawson examples, with stabilizer SO(2), the austere SL 3-folds, withstabilizer S3, the asymptotically conical SL 3-folds, with stabilizer Z3

and the Lawlor-Harvey-Joyce examples, with stabilizer Z2.This present work extends these results to dimension m = 4, namely

tries to classify the special Lagrangian 4-folds in C4 whose fundamentalcubic at a generic point has nontrivial SO(4)-stabilizer.

The possible stabilizer of a traceless cubic can be a continuous,meaning a positive dimensional, or a discrete subgroup of SO(4). InChapter 3.2, we consider the case when the stabilizer is continuous. Itturns out that there are four cases when there are nontrivial specialLagrangian 4-folds with continuous stabilizer type:

(a) When the fundamental cubic has stabilizer SO(3) we obtain theHarvey and Lawson examples which appeared also in dimension3: Lc = (s + it)u | u ∈ S3 ⊂ R4, Im(s + it)4 = c, where c is anyreal constant.

(b) When the stabilizer is SO(2) S3, we obtain special Lagrangiansubmanifolds as products of the form L = R2 × Σ, where Σ ⊂ C2

is a complex curve.

(c) When the stabilizer is SO(2), we obtain the SO(3)-invariant spe-cial Lagrangian 4-folds.

(d) When the stabilizer is an O(2), we obtain a two parameter familyof solutions which we have not been able to integrate completelyyet.

In Chapter 3.3, we consider the case when the stabilizer of the fun-damental cubic is a discrete subgroup of SO(4). In Chapter 3.3.1, weclassify the SL 4-folds with polyhedral stabilizer type. It turns out thatthe polyhedral subgroups of SO(4) that stabilize a traceless cubic in4 variables are the tetrahedral subgroup T, the irreducibly acting oc-tahedral subgroup O+ and the irreducibly acting icosahedral subgroupI+. We show that the special Lagrangian 4-folds whose stabilizer of

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its fundamental cubic is isomorphic to the tetrahedral subgroup are theHarvey-Lawson examples invariant under a torus action, the ones whosestabilizer at a generic point is isomorphic to O+ are the cones on flat3-dimensional tori in the 7-sphere and that there are no nontrivial spe-cial Lagrangian 4-folds whose stabilizer of its fundamental cubic at ageneric point is isomorphic to I+.

Using the classification of the discrete subgroups of SO(4) fromChapter 3.1, it remains to analyze the cases when the stabilizer of thetraceless cubic is a cyclic or a dihedral subgroup of SO(4). We showthat the discrete stabilizer can only have elements of order less or equalto 6. Further, we show that if the stabilizer is discrete and contains anelement of order 6, 5 or 4, then there are no special Lagrangian 4-foldsin C4 with a cyclic or dihedral stabilizer type.

When the stabilizer contains an element of order 3, there are twoinequivalent orbits in the space of fixed traceless cubics that have tobe considered. In the first case of discrete stabilizer type at least a Z3,the special Lagrangian 4-folds whose cubic stabilizer at a generic pointis isomorphic to D3, the dihedral group in 3 elements, turn out to beasymptotically conical. The SL 4-folds with stabilizer type an order 18normal subgroup of D3 × D3 turn out to be products of holomorphiccurves. When the stabilizer type is exactly a Z3, we were able to showthat there is an infinite parameter family of solutions that depends on4 functions of 1 variable, foliated by minimal Legendrian surfaces andby holomorphic curves, but could not finalize the analysis and describethis family completely. In the second case of discrete stabilizer type atleast Z3, we found a large family of SL 4-folds defined by holomorphicdifferential equations, a family that did not appear in dimension 3.

The general case of discrete stabilizer type at least a Z2 is the mostcomplicated case, since the general traceless cubic has a large numberof parameters, and was not considered in this work.

Acknowledgements. I would like to thank my advisor Prof.Robert Bryant for introducing me to the subject, for his help and sup-port and for the innumerable hours of discussions that led to this work.

2. Special Lagrangian geometry in Calabi-Yau manifolds

2.1 Special Lagrangian submanifolds

We begin with the definition of a Calabi-Yau manifold.


Definition 2.1. A Calabi-Yau m-fold (M, J, g) is a compact, com-plex m-dimensional manifold (M, J) with trivial canonical bundle KM

and Ricci-flat Kahler metric g.

Because the canonical bundle KM is trivial, there is a nonzero holo-morphic section Ω of KM . Since the metric g is Ricci-flat, Ω is a paralleltensor with respect to the Levi-Civita connection ∇g [6]. By rescalingΩ, we can take it to be the holomorphic (m, 0)-form that satisfies:

ωm

m!= (−1)

m(m−1)2

(i

2

)m

Ω ∧ Ω,(2.1)

where ω is the Kahler form of g. The form Ω is called the holomorphicvolume form of the Calabi-Yau manifold M .

The special Lagrangian submanifolds were introduced by Harveyand Lawson in their paper [12] using calibrations. They are defined inthe general setting of a Calabi-Yau manifold and are a special class ofminimal submanifolds.

Definition 2.2. Let (M, J, g,Ω) be a Calabi-Yau m-fold and L ⊂M a real m-dimensional submanifold of M . Then L is called a specialLagrangian submanifold of M if ω |L≡ 0 and Im Ω |L≡ 0.

More generally, L is said to be a special Lagrangian submanifoldwith phase eiθ if ω |L≡ 0 and Im(eiθΩ) |L≡ 0.

As an example, we can see that Rm ⊂ Cm is a special Lagrangiansubspace. Cm is endowed with the standard Calabi-Yau structure de-fined by

g0 = dz1 dz1 + · · · + dzm dzm(2.2)

ω0 =i

2(dz1 ∧ dz1 + · · · + dzm ∧ dzm)(2.3)

Ω0 = dz1 ∧ · · · ∧ dzm(2.4)

where g0 is the Kahler metric on Cm, ω0 the Kahler form and Ω0 theholomorphic volume form on Cm. An m-dimensional submanifold L ofM is called Lagrangian if ω |L= 0. So, the special Lagrangian subman-ifolds of M are the Lagrangian submanifolds with the extra conditionIm Ω |L= 0, which is exactly the reason for their name.

There are some important results on SL submanifolds which we willbriefly recall here.a. Deformations. R. McLean [13] studied the moduli space of com-pact special Lagrangian deformations and showed that it has the fol-lowing description:

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Theorem 2.3 (McLean). Let (M, J, g,Ω) be a Calabi-Yau m-foldand L ⊂ M a m-dimensional compact SL submanifold. Then the modulispace ML of special Lagrangian deformations of L is a smooth manifoldof dimension b1(L), the first Betti number.

b. Local existence. Harvey and Lawson [12] proved local existenceonly for SL-submanifolds in Cm, but their result extends to show that if(M, J, g,Ω) is a Calabi-Yau m-fold and N ⊂ M a real analytic subman-ifold of dimension m − 1 such that i∗(ω) = 0, then N lies in a uniqueirreducible SL submanifold L ⊂ M . Here i : N → M is the inclusionmap.

This result shows that there are many special Lagrangian submani-folds locally.

c. Minimizing property. A closed special Lagrangian submanifoldis volume-minimizing in its homology class and therefore it is a mini-mal submanifold. We remark that a minimal submanifold, i.e., a sub-manifold with constant mean curvature 0, is not necessarily volume-minimizing amongst homologous submanifolds. For example the equa-tor of a 2-dimensional sphere is minimal, but does not minimize lengthamongst lines of latitude.

2.2 Structure equations

a. The coframe bundle. Let (M, J, g,Ω) be a Calabi-Yau m-fold andlet Cm ∼= R2m have complex coordinates (z1, z2, . . . , zm) and complexstructure I. The standard Calabi-Yau structure on Cm is given by therelations (2.2), (2.3) and (2.4). Let π : P → M denote the bundle ofCm-valued Calabi-Yau coframes, i.e., an element of Px = π−1(x) is acomplex linear vector space isomorphism u : TxM → Cm that satisfiesωx = u∗(ω0) and Ωx = u∗(Ω0). Then π : P → M is a principal rightSU(m)-bundle over M and the right action is given by Ra(u) = a−1 ufor a ∈ SU(m). Px is the fiber at x of the Calabi-Yau coframe bundleP .

The canonical form ξ is defined on the Calabi-Yau coframe bundleP by

ξu = u (dπ)u : TuP → Cm for u ∈ P

where (dπ)u : TuP → Tπ(u)M is the differential of π at u. The 1-formξ is Cm-valued and we denote its components by ξi, i = 1 . . . m. Then,


on the bundle P the following equations hold:

π∗(ω) =i

2(ξ1 ∧ ξ1 + · · · + ξm ∧ ξm) and π∗(Ω) = ξ1 ∧ · · · ∧ ξm.(2.5)

By regarding the forms on M embedded into the forms on P via thepullback, we can ignore π∗ in the above equations.

We define also the functions ei : P → TM such that ξi(ej) = δij .So, if v ∈ TuP then: (dπ)u(v) = ei(u)ξi(v). Cartan’s first structureequation:

dξi = −ψi ∧ ξj(2.6)

defines (ψi) = ψ, the su(m)-valued 1-form on P called the connectionform. In the flat case M = Cm, Cartan’s second structure equationsatisfied by the connection form ψ is:

dψ = −ψ ∧ ψ.(2.7)

b. Special Lagrangian submanifolds in Cm. In this paper we areinterested in special Lagrangian submanifolds of Cm, therefore we areconsidering only the flat case from now on. When M = Cm with thestandard Calabi-Yau structure (Cm, J, g0, Ω0), we denote the Calabi-Yau coframe bundle by x: P → Cm and regard the functions ei asvector-valued functions on P ∼= Cm × SU(m). Then the relations:

dx = eiξi(2.8)dei = ejψjı(2.9)

give the 1-forms ξi, ψi which form a basis for the space of 1-forms onthe frame bundle P .

To study the SL submanifolds of Cm, we separate the two structureequations (2.6) and (2.7) into real and imaginary parts. We set ξi =ωi + iηi and ψi = αij + iβij . The first structure equation (2.6) becomes

dωi = −αij ∧ ωj + βij ∧ ηj and dηi = −βij ∧ ωj − αij ∧ ηj(2.10)

where we used Einstein’s convention to sum over repeated indices. Sinceψ is skew-hermitian with trace 0, it follows that α = (αij) is skew-symmetric and β = (βij) is symmetric with vanishing trace.

When split into its real and imaginary parts, the second structureequation (2.7) becomes:

dαij = −αik ∧ αkj + βik ∧ βkj(2.11)dβij = −βik ∧ αkj − αik ∧ βkj .(2.12)

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Let L ⊂ M be a special Lagrangian submanifold. We are going toconsider the bundle PL of L-adapted coframes over L. This is definedas follows: Let x ∈ L. A Calabi-Yau coframe at x, u : TxM → Cm issaid to be L-adapted if u(TxL) = Rm ⊂ Cm and u : TxL → Rm preservesorientation. The space of L-adapted coframes forms a principal rightSO(m)-subbundle PL ⊂ π−1(L) ⊂ P over L. Now, because u takes atangent plane to L in M into a real one, ξ is Rm-valued on PL and soηi = 0 holds on PL. By the structure equation (2.10), we get that:

dωi = −αij ∧ ωj and βij ∧ ωj = 0 on PL.

Since ω1, . . . , ωm are linearly independent forms and βij ∧ ωj = 0, Car-tan’s Lemma implies that βij = hijkωk where hijk = hikj . Since βij

is symmetric, hijk = hjik also holds and so hijk are fully symmetricfunctions on the bundle PL.c. The fundamental cubic. Let L ⊂ M = Cm be a SL submanifoldand let ν → L be the normal bundle of L in M , such that TM |L=TL ⊕ ν. The second fundamental form of L is a quadratic form withvalues in the normal bundle ν and it can be interpreted as a tracelesssymmetric cubic form in the following way: The second fundamentalform B of L in M can be written as B = Jei ⊗ hijkωjωk, where hijk

are the fully symmetric functions determined by βij as described above.All the information of the second fundamental form is contained in thesymmetric cubic form C = hijkωiωjωk which is called the fundamentalcubic of the special Lagrangian submanifold L. We note that this cubicis traceless with respect to the induced metric on L, g = ω2

1 + · · ·+ ω2m,

since:trgC = hiikωk = βii = 0.

The following result tells us that the necessary and sufficient con-ditions for the existence of a special Lagrangian in Cm with a givenmetric and a given fundamental cubic are the Gauss and Codazzi-typeequations (2.11) and (2.12).

Let (L, g) ⊂ Cm be a simply connected Riemannian manifold ofdimension m and C a symmetric cubic which is traceless with respect tog. Choose a g-orthonormal coframing ω = (ωi) on an open neighborhoodU ⊂ L and define ηi = 0. Now, let αij = −αji be the unique 1-formson U s.t. dωi = −αij ∧ ωj . Write the cubic as C = hijkωiωjωk and setβij = hijkωk .

Theorem 2.4 (see [1]). Suppose that the forms βij determined byC together with the forms αij determined by (ωi) satisfy the Gauss and


Codazzi equations (2.11), (2.12). Then there is an isometric immersionof (L, g) into Cm as a special Lagrangian submanifold inducing C as itsfundamental cubic. Moreover, this isometric immersion is unique up torigid motions.

3. Second order families

3.1 Discrete subgroups of SO(4)

As we have seen in Section 2.2.c, the second fundamental form C of aspecial Lagrangian submanifold L ⊂ Cm can be regarded as a symmet-ric cubic form in n variables x1, x2, . . . , xn, with vanishing trace withrespect to the induced metric g. It is easy to see that the symmetriccubic is traceless if and only if it is a harmonic cubic, i.e., ∆C = 0,where ∆ =

∑ni=1

∂2

∂x2i. Therefore, the fundamental cubic of a special La-

grangian submanifold in Cm belongs to the space H3(R4) of harmoniccubics in 4 variables. This space is an irreducible SO(4)-module of di-mension 16 and the action is given by

(A · P )x = P (xA)(3.13)

where

A = (aij) ∈ SO(4), P (x) ∈ H3(R4), x = (x1, x2, x3, x4) ∈ R4

and(xA)i = xjaji

is given by usual matrix multiplication.We want to study the families of special Lagrangian 4-folds in C4

whose fundamental cubic at a generic point has nontrivial SO(4)-stab-ilizer. The stabilizer G of a polynomial P (x) ∈ H3(R4) is defined as

G = A ∈ SO(4)| (A · P )(x) = P (x), for any x ∈ R4.

The stabilizer can be either a positive dimensional subgroup of SO(4)or else a discrete subgroup of SO(4). In our analysis, we need to knowwhich are the discrete subgroups of SO(4) that can stabilize a harmoniccubic in 4 variables.

We start by listing the discrete subgroups of SO(4) not containingthe central rotation −I4, since a subgroup of SO(4) that contains −I4

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cannot stabilize any nontrivial cubic polynomial. For a complete proofof the classification of the discrete subgroups of SO(4) the reader mightwant to consult [11].

In the study of the discrete subgroups of SO(4), we are going to usethe quaternionic field H. Let E4 be the Euclidean 4-dimensional spaceand let 1, i, j,k be an orthonormal basis. We define a multiplicationof elements of E4 by the well-known rules: i2 = j2 = k2 = −1, ij =k, jk = i, ki = j. The elements of E4 form the non-commutativefield of quaternions H. We will denote a quaternion by the ordered set(w, x, y, z) or by w +xi+yj+ zk. For a quaternion q = w +xi+yj+ zkwe define the conjugate q = w − xi − yj − zk and the modulus of q as|q| = (qq)

12 . If |q| = 1, we call q a unit quaternion and U = q ∈ H |

|q| = 1 is a multiplicative group called the group of unit quaternions.For any q ∈ E4, we define the right multiplication map ρq : E4 → E4

by ρq(x) = xq and the left multiplication map λq : E4 → E4 by λq(x) =qx. If u ∈ U , both ρu and λu are seen to be in SO(4) and they are calledthe right rotation and the left rotation, respectively. The right rotationsρu : u ∈ U form a group U+ and the left rotations λu : u ∈ U forma group U− and both U+ and U− are subgroups of SO(4), isomorphic tothe unit quaternions group U . These are different subgroups of SO(4)and we notice that U+ ∩ U− = ±1.

Consider now the homomorphism Φ: U × U → SO(4) with

Φ(u1, u2)(x) = u1xu2 = λu1ρu2(x).

It is well-known that Φ is surjective and its kernel is the 2-element group(1, 1), (−1,−1) ∼= Z2. So, U × U/(1,1),(−1,−1) ∼= SO(4) and to studythe subgroups of SO(4) we would be interested in the subgroups of theunit quaternions group U . We also define the surjective 2:1 homomor-phism Ψ: U → SO(3) by Ψ(u)(x) = uxu, whose kernel is ±1.

It is well-known [14] that the discrete subgroups of SO(3) are thecyclic groups, the dihedral groups and the pure symmetry groups ofthe platonic solids which are the tetrahedral, the octahedral and theicosahedral groups. The tetrahedral group T is the group of rotationalsymmetries of a tetrahedron, it has order 12 and it is isomorphic to A4,the group of even permutations of 4 elements. The octahedral group Ois the group of rotational symmetries of an octahedral (or a cube) andit is a group of order 24, isomorphic to S4, the permutation group of 4elements. The icosahedral group I is the group of rotational symmetriesof a icosahedral (or a dodecahedral) and it has order 60, isomorphic toA5.


Using the homomorphism Ψ, it is not hard to determine the discretesubgroups of the group of unit quaternions U . A complete descriptionof how this is done can be found in [11].

Proposition 3.1. Every finite subgroup of the unit quaternionicgroup U is conjugate to one of the following groups:

Cn =

cos2kπ

n+ k sin

2kπ

n, k = 0, . . . , n − 1

Dn = C2n ∪ iC2n

T =2∪

k=0

(12

+12i +

12j +

12k)k

D2

O = T ∪ 1√2(1 + i)T

I =4∪

k=0

(12τ

+τ

2i +

12j)k

T

where τ =√

5+12 .

The binary polyhedral subgroups T,O and I are called, respectively,the binary tetrahedral, octahedral and icosahedral subgroups of U andare twice the order of the corresponding polyhedral subgroup of SO(3).

Now, to find the discrete subgroups of SO(4) we use the 2 : 1 homo-morphism:

Φ: U × U → SO(4)Φ(l, r)(x) = lxr

with Ker Φ = (1, 1), (−1,−1). We denote by [l, r] = (l, r), (−l,−r).Then we have an isomorphism ϕ : U × U/Z2

∼= SO(4) with ϕ([l, r]) =x → lxr.

If σ ⊂ SO(4) is a discrete subgroup, then

L = l ∈ U |x → lxr ∈ σ and R = r ∈ U |x → lxr ∈ σ

are subgroups of U . We notice that σ ⊆ Φ(L × R) but the equalitymight not hold since it might be possible to find a pair (l, r) ∈ L × Rsuch that x → lxr ∈ σ. We define the subgroups

L′ = l ∈ L|x → lx ∈ σ = l ∈ L|(l, 1) ∈ σR′ = r ∈ R|x → xr ∈ σ = r ∈ R|(1, r) ∈ σ

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and there is an isomorphism between the quotient groups L/L′ and R/R′

given by ψ(lL′) = rR′ such that (l, r) ∈ σ. The subgroup Φ(L′ × R′)is normal in σ and σ/Φ(L′×R′)

∼= L/L′ and R/R′ . The subgroup σdepends on the isomorphism ψ between the quotient groups L/L′ andR/R′ , different isomorphism possibly yielding non-conjugate subgroupsin SO(4). We will denote the group σ by (L/L′ ; R/R′)ψ.

For example, Cm is a normal subgroup of Cmr and the quotientgroup Cmr/Cm

∼= Zr. The elements of Cmr/Cm are the cosets piCm,i = 0..r − 1, where p = cos 2π

mr + k sin 2πmr is a generator of Cmr.

If q = cos 2πnr + k sin 2π

nr is a generator of Cnr, we have the isomor-phism ψs : Cmr/Cm → Cnr/Cn defined by ψs(piCm) = qsiCn, i =0..r − 1. For each s such that (s, r) = 1 and s < 1

2r, we get anisomorphism ψs that gives distinct subgroups (Cmr/Cm ;Cnr/Cn)ψs .The subgroups of SO(4) of this form that do not contain the centralrotation are seen to be (C2mr/Cm ;C2nr/Cn)ψs , of order mnr withm and n odd. Also, extending the isomorphism between C2mr/Cm

and C2nr/Cn to one between Dmr/Cm = (C2mr ⊕ iC2mr)/Cm andDnr/Cn = (C2nr ⊕ iC2nr)/Cn by ψs(ipjCm) = iqsjCn, j = 0...r − 1,we obtain the subgroup (Dmr/Cm;Dnr/Cn)ψs of order 2mnr, wherem and n are odd.

Another subgroup of SO(4), of order 12, that does not contain thecentral rotation is

T = (T/C1;T/C1) = (T;T) = [t, t] | t ∈ T.

In the case case when L = R = O and L′ = R′ = C1, we obtain twonon-conjugate groups, depending on the automorphism of O considered.If we take ψ : O → O to be the identical automorphism we obtain thesubgroup O = (O;O) = [o, o], o ∈ O of order 24. If we considerthe automorphism ψ : O = T ⊕ (1 + 1√

2i)T → O with ψ(o) = o, if

o ∈ T and ψ(o) = −o, if o ∈ 1√2(1+ i)T, we obtain a different subgroup

O+ = [o, o], o ∈ T and [o,−o], o ∈ 1√2(1 + i)T, of order 24.

In the case when L = R = I and L′ = R′ = C1, we obtain againtwo non-conjugate subgroups of SO(4) which do not contain the centralrotation. If we take ψ : I → I to be the identical automorphism, weobtain the subgroup I = (I; I) = [l, l], l ∈ I, of order 60. But wenotice that all the elements of I are in the field of rational numbers over√

5 and the change of sign of√

5 interchanges ±τ with ∓τ−1. If p ∈ I isa quaternion we denote by p+ its image under this automorphism. Then

I is interchanged with a group I+ =4∪

k=0( τ

2 + 12τ i + 1

2 j)kT and the two


groups have in common T. If we now consider the isomorphism ψ : I+ →I, ψ(p+) = p then we obtain a different subgroup I+ = [r+, r], r ∈ I,of order 60. This group leaves no axis fixed and it can be shown to bethe rotational symmetry group of a regular simplex in E4, with verticesat 1 and 1

4(−1 ±√

5i ±√

5j ±√

5k).To conclude, we have the following:

Proposition 3.2. The discrete subgroups of SO(4) that do not con-tain the central symmetry −I4 are the following:

1. (C2mr/Cm ;C2nr/Cn)ψs , of order mnr, where m, n odd ;

2. (Dmr/Cm;Dnr/Cn)ψs , of order 2mnr, where m, n odd ;

3. T = (T/C1;T/C1) = [t, t] | t ∈ T, of order 12;

4. O = (O/C1;O/C1) = [o, o] | o ∈ O, of order 24;

5. O+ = (O/C1;O/C1) = [o, o] | o ∈ T and [o,−o], o ∈ 1√2(1 +

i)T, of order 24;

6. I = (I/C1; I/C1) = [l, l] | l ∈ I, of order 60;

7. I+ = (I+/C1; I|C1) = [r+, r] | r ∈ I, of order 60.

For complete proof of this proposition, the reader should consult [11]and [3].

3.2 Continuous stabilizer type

Any maximal torus in SO(4) is conjugate to the group:(eiθ1 00 eiθ2

), θ1, θ2 ∈ [0, 2π)

.

We are looking to determine the orbits of the action (3.13) that havenontrivial stabilizer under the action of SO(4). In what follows, a pos-itive dimensional stabilizer will be called a continuous stabilizer. Thespecial Lagrangian 4-folds whose fundamental cubic at each point hasstabilizer G, a subgroup of SO(4), will be said to have stabilizer typeG. If the stabilizer G is a continuous or discrete subgroup of SO(4), wewill say that the special Lagrangian 4-fold has continuous or discretestabilizer type, respectively.

First, we are going to classify the harmonic cubic polynomials in4 variables x1, x2, x3, x4 whose stabilizer is a continuous subgroup ofSO(4).

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Proposition 3.3. The SO(4)-stabilizer of h ∈ H3(R4) is a con-tinuous subgroup of SO(4) if and only if h lies on the SO(4)-orbit ofexactly one of the following polynomials:

1. 0 ∈ H3(R4), whose stabilizer is SO(4);

2. rx1(x21 − x2

2 − x23 − x2

4) for some r > 0, whose stabilizer is thesubgroup SO(3), consisting of rotations in the 3-space (x2, x3, x4);

3. r[(x21 − x2

2)x3 + 2x1x2x4], for some r > 0, whose stabilizer is thesubgroup O(2) generated by rotations by an arbitrary angle in the(x1, x2)-plane and twice that angle in the (x3, x4)-plane and the

element(

1 0 0 00 −1 0 00 0 1 00 0 0 −1

);

4. r(x31 − 3x1x

22) for some r > 0, whose stabilizer is the subgroup

SO(2) S3, where S3 is the symmetric group on 3 elements gen-erated by the rotation by an angle of 2π

3 in the (x1, x2)- plane and

the element(

1 0 0 00 −1 0 00 0 −1 00 0 0 1

), and SO(2) is the group of rotations in

the (x3, x4)-plane;

5. r(x31−3x1x

22)+3vx1(x2

1+x22−2x2

3−2x24) for some v > 0 satisfying

r = 3v whose stabilizer is the O(2)-subgroup generated by rotations

in the (x3, x4)-plane and the element(

1 0 0 00 −1 0 00 0 1 00 0 0 −1

);

6. r(x31−3x1x

22)+s(3x2

1x2−x32)+3vx1(x2

1 +x22−2x2

3−2x24) for some

s > 0 and v > 0 whose stabilizer is the SO(2)-subgroup generatedby rotations in the (x3, x4)-plane.

Remark (special values). The case r = 3v of case 5 above reducesto case 2, when the stabilizer is SO(3).

Proof. Suppose h ∈ H3(R4) has a nontrivial stabilizer G ⊆ SO(4).If G = SO(4), then h = 0 since H3(R4) is an irreducible representationof SO(4). We suppose from now on that h = 0. Being a stabilizer, Gis a closed subgroup of SO(4), therefore it is compact and has a finitenumber of components.

We suppose G is not discrete. Then, its identity component His a closed connected subgroup of SO(4). The algebra h of H is asubalgebra of so(4). Using the 2:1 homomorphism Φ: U × U → SO(4),Φ(u1, u2)(x) = u1xu2 from Section 3.1, it is easy to see that so(4) ∼=


so(3)+ ⊕ so(3)−, where so(3)+ and so(3)− are two different copies ofso(3) with intersection the 0 vector. Since dim so(4) = 6, there are thefollowing possibilities for the subalgebra h:

1) dim h = 5. This is not possible for the following reason: h ∩so(3)+ ⊂ so(3)+ is a subalgebra of dimension at least 2 and so(3)+ hasno subalgebras of dimension 2. Therefore, h ∩ so(3)+ = so(3)+ whichimplies that h ⊇ so(3)+. Similarly, it can be shown that h ⊇ so(3)− andit follows from here that h = so(4), which gives a contradiction.

2) dim h = 4. Then h+ = h ∩ so(3)+ is an ideal of dimension atleast 1 in so(3)+ and h− = h ∩ so(3)− is an ideal of dimension at least1 in so(3)−. Consider the projections π± : h → so(3)±. Since Kerπ± = h∓, it follows that Ker π± can have dimension 1 or 3. If Ker π−has dimension 1, π− is onto and Ker π+ has dimension 3. In this case,it follows that h+

∼= so(2)+ and we obtain that h = so(2)+ ⊕ so(3)−. Ifthe dimension of Ker π− is 3, then the dimension of Ker π+ is 1 and weobtain h = so(3)+⊕so(2)−. But so(3)± acts like the group SU(2) on thespace of complexified harmonic cubics in four variables z1, z2, z1, z2,where z1 = x1+ix2, z2 = x3+ix4 and calculations show that this actiondoes not preserve any nontrivial element. Therefore, in this case H doesnot preserve any nontrivial harmonic cubic.

3) dim h = 3. In this case, one can show that, up to conjugacy, theonly possibilities for h are so(3)+, so(3)− and diag(so(3)+ ⊕ so(3)−).But, as discussed in 2) above, so(3)± does not preserve any cubic poly-nomial in 4 variables and consequently we can discard these cases.

We study now the case

h = diag(so(3)+ ⊕ so(3)−) = x+ + x−, x+ ∈ so(3)+, x− ∈ so(3)−.

We can see that, up to conjugacy,

diag(so(3)+ ⊕ so(3)−) =

0 0 0 00 0 −2c 2b0 2c 0 −2a0 −2b 2a 0

, a, b, c ∈ R

.

Therefore, G can be either one of the groups:

G =(

1 00 A

), A ∈ SO(3)

or G =

(det(A) 0

0 A

), A ∈ O(3)

.

We can easily see that the cubic polynomials fixed by the identity com-ponent H are linear combinations of x3

1 and x1(x22+x2

3+x24). It is obvious

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that the only combination that would make the polynomial harmonicis P = rx1(x2

1 − x22 − x2

3 − x24), for some r = 0. One can verify that the

full stabilizer of P is SO(3). By a rotation that reverses the x1-axis, ifnecessary, we can assume that r > 0.

4) dim h = 2. In this case, h = so(2)+ ⊕ so(2)− and H is conjugate

to the maximal torus H =(

eiθ1 00 eiθ2

), θ1, θ2 ∈ [0, 2π)

. It is easy

to see that H does not stabilize any symmetric cubic in 4 variables.

5) dim h = 1. In this case, one can show that the only 1-dimensionalideals in so(4) are conjugate to:

hp,q = (px, qx)| x ∈ so(2), p, q ∈ Z, (p, q) = 1.

It follows that the identity component Hp,q =(

R(pθ) 00 R(qθ)

), θ ∈ R

consists of rotations of angle pθ in the (x1, x2)-plane and of angle qθ inthe (x3, x4)-plane. We are looking for harmonic cubics in x1, x2, x3, x4preserved by Hp,q, for some p and q integers.

Let Vn be the irreducible representation of SO(2) given by rotationsof speed n: eiθ.z = eniθz, where z ∈ C. In our case, the speed prepresentation Vp is given by the action on the (x1, x2)-plane: eiθ.z1 =eipθz1 and Vq is given by the action on the (x3, x4)-plane: eiθ.z2 = eiqθz2,where z1 = x1 + ix2 and z2 = x3 + ix4.

Under the action of Hp,q, the space of symmetric polynomials in 4variables S3(R4) decomposes as:

S3(Vp ⊕Vq) = S3(Vp)⊕ (S2(Vp)⊗S1(Vq))⊕ (S1(Vp)⊗S2(Vq))⊕S3(Vq).

But one can see that S3(Vp) ∼= V3p⊕Vp. A basis in V3p is Re z31 , Im z3

1 =x3

1−3x1x22, 3x2

1x2−x32 and a basis in Vp is Re(z1z1z1), Im(z1z1z1) =

(x21 + x2

2)x1, (x21 + x2

2)x2. Similarly, S2(Vp) ∼= V2p ⊕ V0∼= V2p ⊕ R and

we calculate that:

S3(R4) = (V3p ⊕ Vp) ⊕ ((V2p ⊕ R) ⊗ Vq)⊕ (Vp ⊗ (V2q ⊕ R)) ⊕ (V3q ⊕ Vq)

= V3p ⊕ Vp ⊕ V2p+q ⊕ V2p−q ⊕ Vp+2q ⊕ Vp−2q ⊕ V3q ⊕ Vq

where we used that Vn ⊗ Vm = Vn+m ⊕ Vn−m. This decomposition isirreducible and the action has a fixed vector if and only if one of theV ’s in the above direct sum is V0, the 1-dimensional representation on


which the action is trivial. This implies one of the following possibilitiesfor the values of p and q:

p = 0, q = 0, p − 2q = 0, q − 2p = 0, p + 2q = 0 or 2p + q = 0.

We note that p and q have symmetric roles since the planes (x1, x2) and(x3, x4) can be interchanged by an orthogonal transformation. There-fore, up to conjugation with an element in SO(4), the only possiblecases are: p = 0, q = 2p and q = −2p. In the case q = 2p, the fact that(p, q) = 1 implies that p = 1 and q = 2 and in the case q = −2p, we cantake p = 1, q = −2. The conclusion is that, unless one of these condi-tions is satisfied, the group Hp,q does not preserve any cubic symmetricpolynomial in four variables. But since the stabilizer in SO(4) coincideswith the stabilizer in O(4), then up to conjugacy in O(4), the last twocases are the same. We will study each of these cases separately.

a) p = 1 and q = 2. Then H1,2 =(

eiθ 00 e2iθ

), θ ∈ [0, 2π)

. If

P is a complexified polynomial in the variables z1, z2, z1, z2, fixedby H1,2, it is easy to see that P should lie in V2 ⊗ V1, therefore P =az2

1z2 +bz12z2, a, b ∈ C. Now, P is a real harmonic polynomial if b = a.

So, any real harmonic cubic C preserved by this group is of the form:

C = Re(az21z2) = r[(x2

1 − x22)x3 + 2x1x2x4] + s[2x1x2x3 − (x2

1 − x22)x4],

with r, s ∈ R. If r2 + s2 = 0, by applying a rotation of angle arctan( sr )

in the (x3, x4)-plane, we can assume that s = 0. We can also assumethat r ≥ 0. The conclusion is that all the harmonic cubics in 4 variablesstabilized by H1,2 are on the SO(4)-orbit of the cubic h = r[(x2

1−x22)x3+

2x1x2x4]. The full stabilizer of h can be shown to be the disconnected

2-piece subgroup H1,2 ∪ gH1,2 ⊂ SO(4), where g =

(1 0 0 00 −1 0 00 0 1 00 0 0 −1

). It is

isomorphic to O(2), because O(2) is the only nonabelian 2-componentcompact group of dimension one.

b) If p = 0, we can consider q = 1. In this case, the identity com-

ponent of G is H0,1 =(

I2 00 eiθ

), θ ∈ R

∼= S1. A complexified cubic

polynomial C, fixed by this group, should belong to V0⊗V1, therefore itshould be a linear combination of z1z2z2, z3

1 and z21z1. Calculations show

C is harmonic if and only if it is a linear combination of the harmonicpolynomials z3

1 , z21z1 − 2z1z2z2. Therefore, the fixed real harmonic

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cubic polynomials C in 4 variables are of the form

C = r(x31 − 3x1x

22) + s(3x2

1x2 − x32)

+ vx1(x21 + x2

2 − 2x23 − 2x2

4) + ux2(x21 + x2

2 − 2x23 − 2x2

4),

with r, s, u, v ∈ R. By making a rotation in the (x1, x2)-plane, we cansuppose that u = 0.

It remains now to determine the full stabilizer of the polynomial

r(x31 − 3x1x

22) + s(3x2

1x2 − x32) + vx1(x2

1 + x22 − 2x2

3 − 2x24),

which we denote by G.If s = 0 and v = 0, calculations show that the full stabilizer of h is

just the identity component, so G = SO(2). By making some rotations,if necessary, we may assume that s, v > 0.

If s = 0, v = 0, and r = 3v, then h = r(x31 − 3x1x

22) + vx1(x2

1 + x22 −

2x23 − 2x2

4) and the full stabilizer is isomorphic to an O(2)-subgroup,because we are also allowed to flip the signs of x2 and x4. In this casewe can suppose that v > 0. In the case s = 0 and r = 3v, h becomes6vx1(x2

1 − x22 − x2

3 − x24) and we saw that this polynomial has stabilizer

SO(3).Finally, if s = v = 0 and r > 0, the polynomial h = r(x3

1 − 3x1x22)

is preserved by the identity component S1, but we can see that it isalso fixed by the element of order 3, A =

(e

2πi3 00 I2

)and by the element

of order 2, g =

(1 0 0 00 −1 0 00 0 −1 00 0 0 1

)and both A and g do not belong to the

identity component. The rotation A and the reflection g form a groupisomorphic to S3, the symmetric group in 3 elements. S3 acts on theidentity component S1 by conjugation and it is easy to compute thatthe action of A on S1 is trivial, while g acts by flipping the circle S1,

namely g.

(I2 00 eiθ

)=

(I2 00 e−iθ

). Using the exact sequence

0 → S1 → G → S3 → 0

one can verify that G = S1 S3. This completes the proof of Proposi-tion 3.3. q.e.d.

In the next step, we are going to analyze each of the cases givenby Proposition 3.3 and classify the SL 4-folds in C4 whose fundamentalcubic form has a continuous symmetry at each point.


3.2.1 SO(3)-symmetry

Example 1. In their paper [12], Harvey and Lawson found thefollowing special Lagrangian submanifolds of C4, invariant under thediagonal action of SO(4) on C4 = R4 × R4:

Lc = (s + it)u | u ∈ S3 ⊂ R4, Im(s + it)4 = c,(3.14)

where c ∈ R is a constant. The variety L0 is an union of four specialLagrangian 4-planes and when c = 0, each component of Lc is diffeo-morphic to R × S3 and it is asymptotic to one pair of 4-planes in L0.

Theorem 3.4. If L ⊂ C4 is a connected nontrivial special La-grangian submanifold whose cubic fundamental form has an SO(3)-sym-metry at each point, then L is, up to rigid motion, an open subset ofone of the Harvey-Lawson examples.

Proof. In the above hypotheses, a trivial special Lagrangian sub-manifold is a special Lagrangian 4-plane. We can see that the funda-mental cubic C of L lies on the orbit of the 0 cubic if and only L istrivial. Therefore, assume the fundamental cubic is not identically van-ishing. The locus where C vanishes is a proper real-analytic subset ofL, so its complement L∗ is open and dense in L. By replacing L byits component L∗, we can assume without loss of generality that C isnowhere vanishing on L. Since the stabilizer of Cx is SO(3) for all x ∈ L,Proposition 3.3 implies the existence of a positive real-analytic functionr : L → R+ with the property that the equation

C = 3rω1(ω21 − ω2

2 − ω23 − ω2

4)

defines an SO(3)-subbundle F of the bundle PL of L-adapted coframes.On the subbundle F , the following identities hold:

β11 β12 β13 β14

β21 β22 β23 β24

β31 β32 β33 β34

β41 β42 β43 β44

=

3rω1 −rω2 −rω3 −rω4

−rω2 −rω1 0 0−rω3 0 −rω1 0−rω4 0 0 −rω1

.(3.15)

Because F is an SO(3)-bundle, the forms α21, a31 and α41 vanish modω1, ω2, ω3, ω4, meaning that there are functions tij on F such that:

α21 = t21ω1 + t22ω2 + t23ω3 + t24ω4(3.16)α31 = t31ω1 + t32ω2 + t33ω3 + t34ω4

α41 = t41ω1 + t42ω2 + t43ω3 + t44ω4.

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Also, there exist functions ri, i = 1, 2, 3, 4 on F such that:

dr =4∑

i=1

riωi.(3.17)

Substituting the relations (3.15), (3.16) and (3.17) into the identities

dβij = −βik ∧ αkj − αik ∧ βkj(3.18)

and using the identities dωi = −αij ∧ ωj , one gets polynomial relationsamong ri, tij which can be solved, leading to relations of the form:

α21 = tω2, α31 = tω3, α41 = tω4, dr = −5rtω1(3.19)

where we denoted t22 = t33 = t44 by t.Differentiating the last equation in (3.19), we get 0 = d(dr) =

−5rd(t) ∧ ω1, implying that there exits a function u on F such that

dt = uω1.(3.20)

Substituting the relations (3.19) and (3.20) into the identities

dαij = −αik ∧ αkj + βik ∧ βkj(3.21)

and expanding out using the identities dωi = −αij ∧ ωj implies therelations:

u = 4r2 − t2

dα32 = α43 ∧ α42 + (t2 + r2)ω3 ∧ ω2

dα42 = a43 ∧ α32 + (t2 + r2)ω4 ∧ ω2

dα43 = α42 ∧ α32 + (t2 + r2)ω4 ∧ ω3.

Differentiating the last equations yields only identities.So, F → L is a SO(3)-bundle on which the 1-forms ω1, ω2, ω3, ω4,

α32, α42, α43 form a basis and they satisfy the structure equations:


dω1 = 0(3.22)dω2 = tω1 ∧ ω2 + α32 ∧ ω3 + α42 ∧ ω4

dω3 = tω1 ∧ ω3 − α32 ∧ ω2 + α43 ∧ ω4

dω4 = tω1 ∧ ω4 − α42 ∧ ω2 − α43 ∧ ω3

dα32 = α43 ∧ α42 + (t2 + r2)ω3 ∧ ω2

dα42 = −α43 ∧ α32 + (t2 + r2)ω4 ∧ ω2

dα43 = α42 ∧ α32 + (t2 + r2)ω4 ∧ ω3

dr = −5rtω1

dt = (4r2 − t2)ω1

and the exterior derivatives of these equations are identities.The last two equations in (3.22) imply that

dr

−5rt=

dt

4r2 − t2= ω1.

This yields d(r85 + t2r−

25 ) = 0 and since L and F are connected, there

exists a function θ on L with |θ| < π8 such that:

r45 = c

45 cos 4θ

r−15 t = c

45 sin 4θ.

From these last two equations and from last equation in (3.22), it followsthat

ω1 =dθ

c(cos 4θ)54

.

Now, setting ηi = c(cos 4θ)1/4ωi for i = 2, 3 and 4 yields:

dη2 = −α23 ∧ η3 − α24 ∧ η4

dη3 = −α32 ∧ η2 − α34 ∧ η4

dη4 = −α42 ∧ η2 − α43 ∧ η3

dα32 = −α34 ∧ α42 + η3 ∧ η2

dα42 = −α43 ∧ α32 + η4 ∧ η2

dα43 = −α42 ∧ α23 + η4 ∧ η3.

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The above equations represent the structure equations of the metric ofconstant curvature 1 on the 3-sphere S3.

Conversely, if dσ2 is the metric of constant curvature 1 on S3, then,on the product L = (−π

8 , π8 ) × S3, consider the quadratic form

g =dθ2 + cos2 4θdσ2

c2(cos 4θ)5/2

and the cubic form

C = 3cos2 4θdσ2dθ − dθ3

c2(cos 4θ)5/2.

The pair (g, C) satisfies the Gauss and Codazzi equations and byTheorem 2.4 this implies that (L, g) can be isometrically immersed as aspecial Lagrangian 4-fold in C4 inducing C as its fundamental cubic. Foreach value of c, there exists a unique corresponding special Lagrangian4-fold. Since the structure equations (3.22) have an SO(4)-symmetryand Harvey and Lawson [12] found that all the special Lagrangian 4-folds in C4, invariant under the diagonal action of SO(4), can be writtenexplicitly as (3.14), the conclusion of the theorem follows. q.e.d.

3.2.2 O(2)-symmetry

According to Proposition 3.3, there are two cases of O(2)-symmetry.The first one gives the following:

Theorem 3.5. There is no connected nontrivial special Lagrangian4-fold in C4 whose cubic fundamental form has an O(2)-symmetry at

each point, where O(2) is the subgroup S1∪g0S1 with S1 =

(eiθ 00 e2iθ

),

θ ∈ R

and g0 =

(1 0 0 00 −1 0 00 0 1 00 0 0 −1

).

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. Proposi-tion 3.3 implies that there exists a function r : L → R+ for which theequation

C = 3r[(ω21 − ω2

2)ω3 + 2ω1ω2ω4]


defines an O(2)-subbundle F ⊂ PL of the L-adapted coframe bundlePL → L. On the subbundle F , the following identities hold:

β11 β12 β13 β14

β21 β22 β23 β24

β31 β32 β33 β34

β41 β42 β43 β44

=

rω3 rω4 rω1 rω2

rω4 −rω3 −rω2 rω1

rω1 −rω2 0 0rω2 rω1 0 0

.(3.23)

Since F is an O(2)-bundle, the following relations hold: α31 = α41 =α32 = α42 = α43 − 2α21 ≡ 0 mod ω1, ω2, ω3, ω4, meaning that thereexist functions tij on F such that:

α42 = t11ω1 + t12ω2 + t13ω3 + t14ω4(3.24)α32 = t21ω1 + t22ω2 + t23ω3 + t24ω4

α31 = t31ω1 + t32ω2 + t33ω3 + t34ω4

α41 = t41ω1 + t42ω2 + t43ω3 + t44ω4

α43 − 2α21 = t51ω1 + t52ω2 + t53ω3 + t54ω4.

Moreover, there exist functions ri on F , i = 1, 2, 3 and 4 such that

dr =4∑

i=1

riωi.(3.25)

Substituting the relations (3.23), (3.24) and (3.25) into the identities

dβij = −βik ∧ αkj − αik ∧ βkj(3.26)

and using the identities dωi = −αij ∧ ωj , one gets polynomial relationsamong ri, tij which can be solved, leading to relations of the form:

α31 = α41 = α32 = α42 = α43 − 2α21 = 0, dr = 0.(3.27)

Substituting (3.23) and (3.27) into the identities dαij = −αik ∧ αkj +βik ∧ βkj yields r = 0, contrary to the hypothesis. q.e.d.

The second case of symmetry O(2) yields the following partial result:

Proposition 3.6. There is a 2-parameter family of connected spe-cial Lagrangian 4-folds with the property that the symmetry group ofthe fundamental cubic at each point is isomorphic to O(2), where O(2)

is the subgroup S1 ∪ g0S1 with S1 =

(I2 00 eiθ

), θ ∈ R

and g0 =(

1 0 0 00 −1 0 00 0 1 00 0 0 −1

).

234 m. ionel

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. Proposi-tion 3.3 implies that there exist functions r, v : L → R, v ≥ 0, r = 3vand an O(2)-subbundle F ⊂ PL over L on which the following identityholds:

C = r(ω31 − 3ω1ω

22) + 3vω1(ω2

1 + ω22 − 2ω2

3 − 2ω24).(3.28)

Since F is an O(2)-bundle, the following relations hold: α21 = α31 =α42 = α32 = α42 ≡ 0 mod ω1, ω2, ω3, ω4 and doing the differentialanalysis as in the previous cases we obtain the following structure equa-tions on the subbundle F :

dω1 = (r − 3v)t2ω1 ∧ ω2(3.29)dω2 = (r − v)t1ω1 ∧ ω2

dω3 = 2vt1ω1 ∧ ω3 + 2vt2ω2 ∧ ω3 + α43 ∧ ω4

dω4 = 2vt1ω1 ∧ ω4 + 2vt2ω2 ∧ ω4 − α43 ∧ ω3

d(α43) = 4v2(t21 + t22 + 1)ω4 ∧ ω3

dr = −t1(3r2 − rv + 6v2)ω1 + t2(r − 3v)(3r − 2v)ω2

dv = −t1v(7v + r)ω1 + t2v(r − 3v)ω2

d(t1) = [r(t21 + t22 + 1) + v(5t21 − 3t22 + 5)]ω1

d(t2) = 8vt1t2ω1 + (v − r)(t21 + t22 + 1)ω2

for some functions t1, t2. Differentiating these equations yields onlyidentities.

We were not able to integrate completely the structure equationsand find the family of special Lagrangian 4-folds that are solutions tothese equations. One thing we could observe is that the generic solutionhas rank 2, since the following relations hold between the parametersr, v, t1 and t2:

d

((t21 + t22 + 1)v

45 (r − 3v)

(r − v)35

)= 0 and

d

((t22(r − 3v) + (r − v)(t22 + 1))v

75

(r − v)45

)= 0.

The symmetry group of the solutions is three dimensional and it is eitherSO(3) or SO(2)×R2, depending on the values of the parameters r, v, t1and t2.


Remark. In principle, the structure equations can be integratedusing the reduction process for special Lagrangian submanifolds withsymmetries, by solving the ODE associated to it, as in [7]. The solutionwould be in terms of the Jacobi elliptic functions. As for now, we donot have an explicit integral yet.

3.2.3 SO(2) S3-symmetry

Theorem 3.7. Suppose that L ⊂ C4 is a connected special La-grangian 4-fold with the property that its fundamental cubic at eachpoint has an SO(2) S3-symmetry. Then L is congruent to a productΣ × R2, where Σ ⊂ C2 is a holomorphic curve.


C = r(ω31 − 3ω1ω

22)

defines an SO(2) S3-subbundle F ⊂ PL of the L-adapted coframebundle PL → L, subbundle on which the 1-forms ω1, ω2, ω3, ω4 and α43

form a basis. Similar calculations as in previous cases show that thestructure equations on F are:

dω1 = t1ω1 ∧ ω2, dω2 = t2ω1 ∧ ω2, dω3 = α43 ∧ ω4,(3.30)dω4 = −α43 ∧ ω3, dα43 = 0dr = −3rt2ω1 + 3rt1ω2

dt1 = −u2ω1 + (t21 + t22 − 2r2 + u1)ω2

dt2 = u1ω1 + u2ω2

for some functions u1, u2. Differentiation of these equations does notlead to new relations among the quantities because the system becomesinvolutive, according to Cartan-Kahler Theorem. This is seen by com-puting the Cartan characters: s1 = 2, s2 = s3 = s4 = 0 and noticingthat the space of integral elements at each point is parametrized by 2parameters u1, u2.

We are looking to integrate the above equations and find the familyof special Lagrangian 4-folds that satisfy the hypothesis of the theorem.The Cartan-Kahler analysis tells us that the solution should depend on2 functions of one variable.

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From the above structure equations, we can see that ω1 = ω2 = 0 andω3 = ω4 = 0 define integrable 2-plane fields on L. The 2-dimensionalleaves of the 2-plane field Γ1 defined by ω3 = ω4 = 0 are congruentalong Γ2, the codimension 2 foliation defined by ω1 = ω2 = 0. Thisis clear since dt1 = dt2 ≡ 0 mod ω1, ω2 and therefore the structureequations of Γ1 are:

dω1 = t1ω1 ∧ ω2, dω2 = t2ω1 ∧ ω2

where t1, t2 are constant along Γ2. Also, the third to fifth equationsin (3.30) imply that the leaves of the foliation Γ2 are 2-planes whichare congruent along Γ1 since d(e3 ∧ e4) = 0 and the 2-planes are real,spanned by e3, e4. Therefore, L is a product L = Σ×R2 where Σ ⊂ C2

is a surface. In order for L to be a special Lagrangian 4-fold, Σ should bea holomorphic curve with respect to a certain unique complex structureon C2. This is because of the following argument: Choose coordinateszk = xk + iyk, k = 1...4 on L = Σ×R2. Then L is special Lagrangian ifand only if the 2-forms dx1 ∧ dy1 + dx2 ∧ dy2 and dx1 ∧ dy2 + dy1 ∧ dx2

each vanish when pulled back to Σ. But

(dx1 ∧ dy1 + dx2 ∧ dy2) + i(dx1 ∧ dy2 + dy1 ∧ dx2)= (dx1 − idx2) ∧ (dy1 + idy2) = du ∧ dv

where u = x1 − ix2 and v = y1 + iy2 are a different set of complexcoordinates on C2. Then Σ ⊂ C2 is special Lagrangian if and only ifdu ∧ dv |Σ= 0, which says that Σ is a holomorphic curve in C2 withrespect to the complex coordinates (u, v) on C2. q.e.d.

3.2.4 SO(2)-symmetry

Theorem 3.8. Suppose that L ⊂ C4 is a connected special La-grangian 4-fold with the property that its fundamental cubic has anSO(2)-symmetry at each point. Then L is invariant under an SO(3)-action, whose orbits are 2-spheres, and the surface we obtain in the quo-tient M of C4 by this action is a pseudo-holomorphic curve with respectto a natural almost complex structure on this quotient M . Conversely,the pre-image of any pseudo-holomorphic curve in M gives a specialLagrangian 4-fold whose fundamental cubic has an SO(2)-symmetry ateach point.


Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. Proposi-tion 3.3 implies that there exist functions r : L → R, s : L → R+ andv : L → R+ for which the equation

C = r(ω31 − 3ω1ω

22) + s(3ω2

1ω2 − ω32) + 3vω1(ω2

1 + ω22 − 2ω2

3 − 2ω24)

defines an SO(2)-subbundle F ⊂ PL of the L-adapted coframe bundlePL → L. The 1-forms ω1, ω2, ω3, ω4 and α43 form a basis and satisfy thestructure equations:

dω1 = [−t1s + (r − 3v)t2]ω1 ∧ ω2, dω2 = [t2s + (r − v)t1]ω1 ∧ ω2

(3.31)

dω3 = 2t1vω1 ∧ ω3 + 2t2vω2 ∧ ω3 + α43 ∧ ω4,

dω4 = 2t1vω1 ∧ ω4 + 2t2vω2 ∧ ω4 − α43 ∧ ω3

dα43 = −4v2(t21 + t22 + 1)ω3 ∧ ω4

dr = [t1(−6v2 + vr − 3s2 − 3r2) − 11t2vs − t4]ω1

+ [−t1vs + t2(6v2 − 11vr + 3r2 + 3s2) + t3]ω2

ds = t3ω1 + t4ω2

dv = −v[t1(r + 7v) + t2s]ω1 + v[−t1s + (r − 3v)t2]ω2

dt1 = [t21(r + 5v) + t22(r − 3v) + r + 5v]ω1 + [s(t21 + t22) + s]ω2

dt2 = [8vt1t2 + s + s(t21 + t22)]ω1 + (v − r)[(t21 + t22) + 1]ω2

dt3 = u1ω1 + [u2 + t2t3(r − 3v) + t1t4(r − v) − t1t3s + t2t4s]ω2

dt4 = u2ω1 + (−u1 − 9t21s3 − 6s3 + 24rsv − 6r2s − 18v2s

− 60t1t2vs2 + 90rsvt22 + 3vt1t3 + 30t21vrs

− 9t22s3 − 7rt1t3 − 9t22r

2s − 21t21v2s − 7t2t3s + 7t2t4r

− 25t2t4v − 141t22v2s − 9t21r

2s − 7t1t4s)ω2

for some functions u1, u2. The above system is in involution, so differ-entiation of these equations does not lead to new relationships amongquantities.

From the above structure equations, we can see that ω1 = ω2 = 0 andω3 = ω4 = 0 define integrable 2-plane fields on L. The 2-dimensionalleaves of the 2-plane field Γ1 defined by ω1 = ω2 = 0 are 2-spheres. Thisis clear since the structure equations of the leaves of Γ1 are the structureequations of a 2-dimensional sphere of constant radius 4v2(t21 + t22 + 1):

dω3 = −α34 ∧ ω4, dω4 = α34 ∧ ω3, dα34 = 4v2(t21 + t22 + 1)ω3 ∧ ω4

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and t1, t2, v are constant along Γ1 since dt1≡dt2≡dv≡0 mod ω1, ω2.Therefore L is foliated by non-congruent spheres.

The 2-dimensional leaves of the other foliation Γ2, defined by ω3 =ω4 = 0, are congruent. This follows from the structure equations

dω1 = [−t1s + (r − 3v)t2]ω1 ∧ ω2, dω2 = [t2s + (r − v)t1]ω1 ∧ ω2

and the fact that dr ≡ dv ≡ dt1 ≡ dt2 ≡ 0 mod ω1, ω2.Also, the structure equations imply d(e1 ∧ e2 ∧ Je1 ∧ Je2) = 0

mod ω3, ω4. Therefore the complex 2-plane (e1, e2, Je1, Je2) is con-stant along each leaf of the Γ2-foliation and each such leaf lies in anaffine plane parallel to this 2-plane.

If we let ω21 = [−t1s+(r−3v)t2]ω1+[t2s+(r−v)t1]ω2, the structureequation for the Γ2 leaves can be written as:

dω1 = ω21 ∧ ω2, dω2 = −ω21 ∧ ω1, dω21 = 2(r2 + s2 − v2)ω1 ∧ ω2.

This shows that the leaves of the Γ2 foliation are congruent surfacesof Gauss curvature 2(v2 − r2 − s2), lying in the affine complex 2-plane(e1, e2, Je1, Je2).

Computations show that the structure equations are invariant un-der an SO(3)-rotation about some point in C4. Therefore, the solutionsshould be special Lagrangian 4-folds that are invariant under the sub-group SO(3), as it sits naturally in SO(4) and hence in SU(4). Theorbits of the SO(3)-action are 2-spheres.

We look now for special Lagrangian 4-folds L, invariant under theaction of SO(3). Let

z =(

x + iyx4 + iy4

), x = (x1, x2, x3),y = (y1, y2, y3) ∈ R3

denote the coordinates on C4. The subgroup SO(3) acts diagonally byrotation in x and y,

A ·(

x + iyx4 + iy4

)=

(Ax + iAyx4 + iy4

), A ∈ SO(3).

Let X1, X2, X3 be the infinitesimal generators of SO(3), where

X1 = x2∂

∂x3− x3

∂

∂x2+ y2

∂

∂y3− y3

∂

∂y2

X2 = x3∂

∂x1− x1

∂

∂x3+ y3

∂

∂y1− y1

∂

∂y3

X3 = x1∂

∂x2− x2

∂

∂x1+ y1

∂

∂y2− y2

∂

∂y1.


The 4-fold L is invariant under the flow of Xi, i = 1, 2, 3, so (Xiω) |L=0, i = 1, 2, 3, where ω = dx · dy + dx4 ∧ dy4 is the symplectic form and

dx · dy := dx1 ∧ dy1 + dx2 ∧ dy2 + dx3 ∧ dy3.

It is easy to calculate that

(X1ω) |L= d(x2y3 − x3y2)

which implies that x2y3 − x3y2 = c1, where c1 ∈ R is a constant. Simi-larly, we can show that

x3y1 − x1y3 = c2, x1y2 − x2y1 = c3.

From here it follows that

x × y = c = (c1, c2, c3),

where c ∈ R3 is a constant vector.If c = 0, then x,y are linearly independent and therefore the stabi-

lizer of a point on the orbit is trivial. This implies that the orbit hasdimension 3, but we know that the orbits are 2 dimensional spheres. Itfollows that c = 0, i.e., x and y are linearly dependent. So, L lies in the6-manifold Σ ⊂ C4 on which the coordinates are given by

z =(

(x + iy)ux4 + iy4

), x, y ∈ R, u = (u1, u2, u3) ∈ S2.

It is easy to compute that

ω |Σ = d(xu1)∧ d(yu1) + d(xu2)∧ d(yu2) + d(xu3)∧ d(yu3) + dx4 ∧ dy4

= dx ∧ dy + dx4 ∧ dy4.

Dividing out by the SO(3)-action on Σ, we obtain in the quotient a 4-dimensional manifold X4, with coordinates (x, y, x4, y4). The leaves ofthe ω3 = ω4 = 0 foliation are 2-dimensional surfaces M2. We calculatenow the pullback of the volume form to Σ. Denote z = x + iy andw = x4 + iy4. So,

Ω |Σ = dz1 ∧ dz2 ∧ dz3 ∧ dz4 |Σ= d(zu1) ∧ d(zu2) ∧ d(zu3) ∧ dw

= z2(u3du1 ∧ du2 + u1du2 ∧ du3 + u2du3 ∧ du1) ∧ dz ∧ dw

=13d(z3) ∧ dw ∧ dσ

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where dσ = u3du1 ∧ du2 + u1du2 ∧ du3 + u2du3 ∧ du1 is the area form ofthe 2-sphere S2.

Then L ⊂ Σ is special Lagrangian if and only if the 2-forms

α = dx ∧ dy + dx4 ∧ dy4 =i

2(dz ∧ dz + dw ∧ dw)

β = Im(13d(z3) ∧ dw) = − i

6(d(z3) ∧ dw − d(z3) ∧ dw)

each vanish when pulled back to M2 ⊂ X4. But:

α ∧ α = −12(dz ∧ dz ∧ dw ∧ dw)

β ∧ β =118

(d(z3) ∧ dw ∧ d(z3) ∧ dw) =12((zz)2dz ∧ dw ∧ dz ∧ dw).

Rescaling β by dividing it by zz

β = − i

6zz(d(z3) ∧ dw − d(z3) ∧ dw)

we get that (α + iβ)2 = 0, so this form is decomposable and it is easyto compute that

α + iβ =i

2zz(ξ1 ∧ ξ2),

where ξ1 = zdz − izdw and ξ2 = zdz − izdw. The forms ξ1, ξ2 form asystem which is not integrable since

dξ1 =z

zdw ∧ dw = 0 mod ξ1, ξ2.

In fact there is no combination of the forms ξ1, ξ2 that is integrable.Since α + iβ |M= 0, it implies that M2 is a pseudo-holomorphic

curve in X4, with respect to a certain almost complex structure, whichis not integrable. Conversely, every almost complex surface in X4 liftsto a special Lagrangian 4-fold L ⊂ Σ6 ⊂ C4. q.e.d.

3.3 Discrete stabilizer type

Next, we are analyzing the case of discrete stabilizer type. Suppose thatthe stabilizer G of the fundamental cubic of a special Lagrangian 4-foldis a finite subgroup of SO(4). If g is an element of G, then g is conjugateto an element in the maximal torus of SO(4):(

e2πir 00 e2πis

), r ∈ Q, s ∈ Q, r, s < 1

.


The following result tells us when there exists a harmonic cubic in4 variables fixed by a nontrivial element g in the maximal torus:

Proposition 3.9. The element g =(

e2πir 00 e2πis

), where r ∈

Q, s ∈ Q, r, s < 1, fixes a nontrivial harmonic cubic in four variables(x1, x2, x3, x4) if and only if at least one of the following conditions issatisfied:

1. 3r ∈ Z, when the fixed harmonic cubics contain linear combina-tions of x3

1 − 3x1x22, 3x2

1x2 − x32;

2. r ∈ Z, when the fixed harmonic cubics contain linear combinationsof x3

1 −3x1x22, 3x2

1x2 −x32, x1(x2

1 +x22 −2x2

3 −2x24), x2(x2

1 +x22 −

2x23 − 2x2

4);

3. 2r + s ∈ Z, when the fixed harmonic cubics contain linear combi-nations of (x2

1 − x22)x3 − 2x1x2x4, (x2

1 − x22)x4 + 2x1x2x3;

4. 2r − s ∈ Z, when the fixed harmonic cubics contain linear combi-nations of (x2

1 − x22)x3 + 2x1x2x4, (x2

1 − x22)x4 − 2x1x2x3;

5. 2s + r ∈ Z, when the fixed harmonic cubics contain linear combi-nations of (x2

3 − x24)x1 − 2x2x3x4, (x2

3 − x24)x2 + 2x1x3x4;

6. 2s − r ∈ Z, when the fixed harmonic cubics contain linear combi-nations of (x2

3 − x24)x1 + 2x2x3x4, (x2

3 − x24)x2 − 2x1x3x4;

7. 3s ∈ Z, when the fixed harmonic cubics contain linear combina-tions of x3

3 − 3x3x24, 3x2

3x4 − x34;

8. s ∈ Z, when the fixed harmonic cubics contain linear combinationsof x3

3 − 3x3x24, 3x2

3x4 − x34, x3(2x2

1 + 2x22 − x2

3 − x24), x4(2x2

1 +2x2

2 − x23 − x2

4).

Proof. Let P3C = P3(z1, z2, z1, z2) be the space of complexified cubic

polynomials, in the variables (z1, z2, z1, z2), where z1 = x1 + ix2, z2 =x3 + ix4. The maximal torus in SO(4) acts on H3

C = H3(z1, z2, z1, z2),the space of complexified harmonic cubics in 4 variables (z1, z2, z1, z2),as follows:(

e2πir 00 e2πis

).P (z1, z2, z1, z2) = P (z∗1 , z

∗2 , z

∗1 , z

∗2), P ∈ H3

C

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where z∗1 = e2πirz1, z∗2 = e2πisz2. Under this action, the space H3C

decomposes as follows:

H3C = H3(z1, z2) ⊕H(P2(z1, z2) ⊗ P1(z1, z2))

⊕H(P1(z1, z2) ⊗ P2(z1, z2)) ⊕H3(z1, z2).

A basis for the space H3(z1, z2) is given by the polynomials z31 , z

21z2,

z1z22 , z

32. Since g.z1 = e2πirz1 and g.z2 = e2πisz2, it follows that

g.z31 = e6πirz3

1 , g.z21z2 = e2πi(2r+s)z2

1z2, g.z1z22 = e2π(r+2s)z1z

22 and

g.z32 = e6πisz3

2 . This further implies that unless e6πir = 1, e2πi(2r+s) = 1,e2πi(r+2s) = 1 or e6πis = 1, there is no fixed element in the spaceH3(z1, z2). The above conditions are equivalent to 3r ∈ Z, 2r + s ∈ Z,r + 2s ∈ Z or 3s ∈ Z. Therefore, H3(z1, z2) decomposes into the follow-ing four weight spaces:

H3(z1, z2) = V(3,0) ⊕ V(2,1) ⊕ V(1,2) ⊕ V(0,3).

All these weight spaces have multiplicity 1 and a basis in V(3,0), V(2,1),V(1,2), V(0,3) is given by the harmonic polynomials z3

1 , z21z2, z1z

22 and z3

2

respectively.We analyze now the fixed elements for the space H(P2(z1, z2) ⊗

P1(z1, z2)) of harmonic polynomials in P2(z1, z2)⊗P1(z1, z2). It is easyto see that a basis in the space H(P2(z1, z2) ⊗ P1(z1, z2)) is given bythe harmonic cubics

z21z2, z2

1z1 − 2z1z2z2, z22z2 − 2z1z2z1, z2

2z1

and this space decomposes into:

H(P2(z1, z2) ⊗ P1(z1, z2)) = V(2,−1) ⊕ V(1,0) ⊕ V(0,1) ⊕ V(−1,2).

We can see that unless at least one of the conditions: 2r − s ∈ Z, r ∈Z, s ∈ Z or 2s − r ∈ Z are satisfied, there is no fixed vector in any ofthe weight spaces.

Doing a similar argument, one can see that a basis in the spaceH(P1(z1, z2) ⊗ P2(z1, z2)) is given by the polynomials

z2z12, z1z1

2 − 2z2z1z2, z2z22 − 2z1z1z2, z1z2

2

and the space decomposes into:

H(P1(z1, z2) ⊗ P2(z1, z2)) = V(−2,1) ⊕ V(−1,0) ⊕ V(0,−1) ⊕ V(1,−2).


Unless at least one of the conditions: −2r + s ∈ Z,−r ∈ Z,−s ∈ Z or−2s + r ∈ Z is satisfied, there is no fixed vector in any of the weightspaces.

Finally, a basis in the space H3(z1, z2) is given by the polynomialsz1

3, z12z2, z1z2

2, z23 and this space decomposes into the weight spaces:

H3(z1, z2) = V(−3,0) ⊕ V(−2,−1) ⊕ V(−1,−2) ⊕ V(−0,−3).

For there to be a fixed vector in this space, at least one of the followingconditions should be satisfied: −3r ∈ Z, −2r − s ∈ Z, −r − 2s ∈ Z or−3s ∈ Z.

Therefore, the space of complexified harmonic cubics decomposesunder the action of the maximal torus into 8 pairs of opposite weightspaces, each of multiplicity one:

H3C = V(3,0) ⊕ V(−3,0) ⊕ V(2,1) ⊕ V(−2,−1) ⊕ V(1,2)

⊕ V(−1,−2) ⊕ V(0,3) ⊕ V(0,−3) ⊕ V(2,−1)

⊕ V(−2,1) ⊕ V(1,0) ⊕ V(−1,0) ⊕ V(0,1) ⊕ V(0,−1) ⊕ V(1,−2) ⊕ V(−1,2).

A real harmonic cubic is the sum of elements drawn from theseweight spaces, with the coefficients in opposite weight spaces being com-plex conjugates. Then, there exists a fixed element in the space of realharmonic cubics in 4 variables if and only if there are nontrivial ele-ments in the maximal torus that act trivially on at least one pair ofthese weight spaces. By the above analysis, one can see that this isequivalent to the satisfaction of at least one of the following conditions:(1) 3r ∈ Z, (2) r ∈ Z, (3) 2r + s ∈ Z, (4) 2r − s ∈ Z, (5) 2s + r ∈ Z, (6)2s − r ∈ Z, (7) 3s ∈ Z, (8) s ∈ Z. Next we assume that exactly one ofthe conditions above is satisfied:

1) 3r ∈ Z. In this case g acts trivially on the pair of opposite weightspaces V(3,0) and V(−3,0) and the fixed real harmonic cubics in 4 variablesare of the form az3

1 + az13. So,

C = Re(az31),

where a ∈ C. Therefore, a basis in the space of fixed real harmoniccubics is given by the harmonic polynomials x3

1 − 3x1x22, 3x2

1x2 − x32.

2) r ∈ Z. This condition implies also 3r ∈ Z and g acts trivially onthe pairs of opposite weight spaces V(3,0), V(−3,0), V(1,0) and V(−1,0). So,the fixed real harmonic cubics are of the form:

C = Re(az31 + b(z2

1z1 − 2z1z2z2)),

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where a, b ∈ C. Therefore, a basis in the space of fixed real harmoniccubics is:

x31−3x1x

22, 3x2

1x2−x32, x1(x2

1+x22−2x2

3−2x24), x2(x2

1+x22−2x2

3−2x24).

3) 2r + s ∈ Z. Then g acts trivially on the pair of opposite weightspaces V(2,1) and V(−2,−1) and the fixed real harmonic cubics are of theform:

C = Re(az21z2),

where a ∈ C. A basis in the space of fixed real harmonic cubics is givenby the polynomials: (x2

1 − x22)x3 − 2x1x2x4, (x2

1 − x22)x4 + 2x1x2x3

4) 2r − s ∈ Z. The element g acts trivially on V(2,−1) and V(−2,1)

and the fixed real harmonic cubics are of the form:

C = Re(az21z2),

where a ∈ C. Thus, a basis in the space of fixed real harmonic cubics isgiven by the polynomials: (x2

1−x22)x3+2x1x2x4, (x2

1−x22)x4−2x1x2x3

5) 2s + r ∈ Z. In this case g acts trivially V(1,2) and V(−1,−2) and

C = Re(az1z22),

where a ∈ C is the general harmonic cubic polynomial fixed by theaction. Therefore, a basis for the space of fixed real harmonic cubicsis given by the harmonic polynomials (x2

3 − x24)x1 − 2x2x3x4, (x2

3 −x2

4)x2 + 2x1x3x4.6) 2s − r ∈ Z. Then g acts trivially on the pair of opposite weight

spaces V(−1,2) and V(1,−2) and the fixed real harmonic cubics are of theform:

C = Re(z22z1),

where a ∈ C. A basis is given by the polynomials: (x23 − x2

4)x1 +2x2x3x4, (x2

3 − x24)x2 − 2x1x3x4

7) 3s ∈ Z. Now g acts trivially on the pair of opposite weight spacesV(0,3) and V(0,−3) and the fixed real harmonic cubics are of the form:

C = Re(az32),

where a ∈ C and therefore, a basis is given by x33−3x3x

24, 3x2

3x4−x34.


8) s ∈ Z. In this last case, g acts trivially on V(0,3), V(0,−3), V(0,1) andV(0,−1). The real harmonic cubics fixed by the action are of the form:

C = Re(az32 + b(z2

2z2 − 2z1z2z1)),

where a, b ∈ C. Therefore, a basis in the space of fixed real harmoniccubics is:

x3

3 − 3x3x24, 3x2

3x4 − x34, x3(2x2

1 + 2x22 − x1

3 − x24),

x4(2x21 + 2x2

2 − x23 − x2

4)

.

q.e.d.

Remark 1. In Figure 1 below we graphed in the coordinates (r, s)mod Z all the possibilities appearing in Proposition 3.9. By modingout by the Weyl group of SO(4), we can consider the possibilities onlyin the triangle found by intersecting the regions below the lines r = sand s = 1− r. Furthermore, since the stabilizer of g in SO(4) coincideswith its stabilizer in O(4), we can actually mod out by the Weyl groupof O(4). The elements

(e2πir 0

0 e2πis

)and

(e2πir 0

0 e−2πis

)are conjugate to

each other in O(4), by the element(

1 0 0 00 1 0 00 0 −1 00 0 0 1

)∈ O(4). Therefore, we

can restrict our attention to the cases found in the small shaded triangleshown in Figure 1, which is a fundamental Weyl chamber.

Remark 2. If only one of the conditions in the small triangle aresatisfied, the stabilizer G will be at least an SO(2), thus continuous, andwe will recover the cases already studied in Section 3.2. For example,if only 2s − r ∈ Z, moding out by Z, we get 2s − r = 0. The stabilizergroup in this case looks like

(e2iθ 00 eiθ

), θ ∈ R

and this is the case of

continuous symmetry studied in Section 3.2.3.

Therefore, in order to find the fundamental cubics that have discretenontrivial stabilizers under the action of SO(4), we have to look atelements that have nontrivial components in at least two non-oppositeweight spaces. As seen in Figure 1, up to conjugacy in O(4), there aresix nontrivial elements in the maximal torus that act trivially on morethan two pairs of weight spaces.

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2r-s=2

2s-r=–1

s=1-r r=s

r+2s=2

r+2s=1

2r+s=22r+s=1 2r-s=12r-s=0

2s-r=1

2s-r=0

1/2

2/3

1/3

1/2 2/31/30

1

s

1r

Figure 1. The weight spaces and a fundamental Weyl chamber.

Corollary 3.10. If G is a nontrivial discrete subgroup of SO(4)that stabilizes a nontrivial polynomial h ∈ H3(R4), then G can not haveelements of order > 6.

Proof. This follows from Proposition 3.9 and the above remarks.Any element in G is conjugate to an element of the form g=

(e2πir 0

0 e2πis

),

where r = mp ∈ Q, s = n

q ∈ Q, r, s < 1.By looking at the small triangle in Figure 1, we can see that there

are the following possibilities for the values of r and s mod Z and modthe Weyl group:

1) If r + 2s ∈ Z and 3r ∈ Z, then r = 23 and s = 1

6 . The element

g =(

e4πi3 0

0 eπi3

)has order 6 and acts trivially on the pairs of opposite

weight spaces V(3,0), V(−3,0) and V(1,2), V(−1,−2). The general harmoniccubic stabilized by this element is

C = Re(az31 + bz1z

22), a, b ∈ C.


2) If r + 2s ∈ Z and 2r − s ∈ Z, then we get r = 35 and s = 1

5 . The

element g =(

e6πi5 0

0 e2πi5

)has order 5 and acts trivially on the pairs

of opposite weight spaces V(1,2), V(−1,−2) and V(2,−1), V(−2,1). Therefore,the general harmonic cubic stabilized by this element is

C = Re(az1z22 + bz2

1z2), a, b ∈ C.

3) If 2s − r ∈ Z and r + 2s ∈ Z, then we get (r, s) = (12 , 1

4). The

element g =(

−1 00 i

)has order 4 and acts trivially on the pairs of

opposite weight spaces V(−1,2), V(1,−2) and V(1,2), V(−1,−2). The generalharmonic cubic stabilized by this element is

C = Re(az1z22 + bz1z

22), a, b ∈ C.

4) If s ∈ Z and 3r ∈ Z, then we get (r, s) = (23 , 0). The element

g =(

e4πi3 00 I2

)has order 3 and acts trivially on the pairs of opposite

weight spaces V(3,0), V(−3,0), V(0,1), V(0,−1) and V(0,3), V(0,−3). The generalharmonic cubic fixed by this element is

C = Re(az31 + bz3

2 + c(z22z2 − 2z1z1z2)), a, b, c ∈ C.

5) If 2s − r ∈ Z, 2r − s ∈ Z, 3r ∈ Z and 3s ∈ Z then we get (r, s) =

(23 , 1

3). The element g =(

e4πi3 0

0 e2πi3

)has order 3 and acts trivially on

the pairs of opposite weight spaces V(−1,2), V(1,−2), V(2,−1), V(−2,1), V(3,0),V(−3,0) and V(0,3), V(0,−3). The general harmonic cubic stabilized by thiselement is

C = Re(az1z22 + bz2

1z2 + cz31 + ez3

2), a, b, c, e ∈ C.

6) If s ∈ Z, 2r + s ∈ Z and 2r − s ∈ Z, then we get (r, s) = (12 , 0).

The element g =(−I2 0

0 I2

)has order 2 and acts trivially on the pairs of

opposite weight spaces V(0,1), V(0,−1), V(2,1), V(−2,−1),V(2,−1), V(−2,1) and V(0,3), V(0,−3). The general harmonic cubic stabilizedby this element is

C = Re(az32 + b(z2

2z2 − 2z1z1z2) + cz21z2 + ez2

1z2), a, b, c, e ∈ C.

7) If r ∈ Z, s ∈ Z, r +2s ∈ Z, 2r + s ∈ Z, 2s− r ∈ Z and 2r− s ∈ Z,meaning all the conditions are satisfied at once, then we get r = 1 ands = 0, so g is just the identity element.

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3.3.1 Polyhedral symmetry

Now we are going to find the nontrivial harmonic cubic polynomialsin 4 variables whose stabilizer is one of the polyhedral subgroups ofSO(4) described in Section 3.1. and we will study the families of specialLagrangian 4-folds with this stabilizer type.

Proposition 3.11. The SO(4)-stabilizer of C ∈ H3(R4) is a poly-hedral subgroup of SO(4) if and only if C lies on the SO(4)-orbit ofexactly one of the following polynomials:

1. rx1(x21 − x2

2 − x23 − x2

4) + sx2x3x4, for some r, s > 0 satisfyings = 2

√5r, whose stabilizer is the tetrahedral subgroup T of SO(4);

2. sx2x3x4, for some s > 0, whose stabilizer is the irreducibly actingoctahedral subgroup O+;

3. r[x1(x21 − x2

2 − x23 − x2

4) + 2√

5x2x3x4], r > 0, whose stabilizer isthe irreducibly acting icosahedral subgroup I+.

Proof. The polyhedral subgroups of SO(4) were found to be thetetrahedral subgroup T of order 12, the reducibly and irreducibly actingoctahedral subgroups O and O+, each of order 24, the reducibly andirreducibly acting icosahedral subgroups I and I+, each of order 60.

First we look at the tetrahedral subgroup T and find the harmoniccubics in 4 variables x1, x2, x3, x4 that are stabilized by this sub-group. As we have seen in Section 3.1, T = [t, t]|t ∈ T, whereT = ±1,±i,±j,±k, 1

2(±1 ± i ± j ± k) is the binary tetrahedral sub-group of the unit quaternion group U , of order 24. The subgroup T

sits in SO(3) and it is generated by the transformations: [i, i] with rep-

resenting matrix[

1 0 0 00 1 0 00 0 −1 00 0 0 −1

], relative to the basis 1, i, j,k, [j, j] with

representing matrix[

1 0 0 00 −1 0 00 0 1 00 0 0 −1

]and [12(1 + i + j + k), 1

2(1 + i + j + k)]

with representing matrix[

1 0 0 00 0 1 00 0 0 10 1 0 0

]. We can see that T fixes a cubic in

x1, x2, x3, x4 if and only if the cubic is invariant under the flips of thesigns of two of the coordinates x2, x3, x4 and also under permutingx2, x3, x4 while keeping x1 fixed. Therefore, the cubic should be alinear combination of the polynomials x3

1, x1(x22 +x2

3 +x24) and x2x3x4.

Now, considering the extra condition that the cubic should be harmonic,it follows that the harmonic cubics stabilized by T lie on the SO(4)-orbit


of the polynomial

C = rx1(x21 − x2

2 − x23 − x2

4) + sx2x3x4,(3.32)

for some r, s ≥ 0.We turn now to the harmonic cubics invariant under the reducibly

acting octahedral subgroup O, which sits in SO(3). As we have seen inSection 3.1, the group O = [o, o] | o ∈ O, where O = T∪ 1√

2(1+i)T is

the octahedral binary subgroup of U , of order 48. Since O contains T, itfollows that O is generated by the generators of T and the extra element[

1√2(1 + i), 1√

2(1 + i)

]with representing matrix

(1 0 0 00 1 0 00 0 0 10 0 −1 0

). This extra

element fixes the polynomial (3.32) if and only if s = 0. Therefore, theharmonic cubics stabilized by O lie on the SO(4)-orbit of the polynomialrx1(x2

1 − x22 − x2

3 − x24), r > 0 which has full symmetry SO(3).

We look now for harmonic cubics invariant under the irreduciblyacting octahedral subgroup O+ = [o, o], o ∈ T and [o,−o], o ∈ 1√

2(1 +

i)T. The subgroup O+ contains T and it is generated by the gener-ators of T plus the extra element

[1√2(1 + i),− 1√

2(1 + i)

]. This extra

element fixes the harmonic polynomial (3.32) if and only if r = 0. There-fore, the harmonic cubics stabilized by O+ lie on the SO(4)-orbit of thepolynomial sx2x3x4, s > 0.

We look now for the harmonic cubics invariant under the reduciblyacting icosahedral subgroup I, which sits in SO(3). As we have seenin Section 3.1, I = [l, l] | l ∈ I, where I = ∪4

k=0(12τ + τ

2 i + 12 j)

kT

is the binary icosahedral subgroup of U , of order 120 and τ =√

5+12 .

The subgroup I contains T and it is generated by the generators ofT plus the extra element

[12τ + τ

2 i + 12 j,

12τ + τ

2 i + 12 j]. Straightforward

calculations show that this extra element fixes the harmonic polynomial(3.32) if and only if s = 0. Therefore, the harmonic cubics stabilized byI lie on the SO(4)-orbit of the polynomial rx1(x2

1 − x22 − x2

3 − x24) which

has full symmetry SO(3).Finally, we look for the harmonic cubics invariant under the ir-

reducibly acting icosahedral subgroup I+. From Section 3.1, I+ =[r+, r] | r ∈ I, where r+ is the image of r ∈ I under the automorphismof the quaternion field that changes the sign of

√5. This automorphism

exchanges τ for − 1τ . The subgroup I+ contains T and it is generated by

the generators of T plus the extra element[( 12τ + τ

2 i+12 j)

+, 12τ + τ

2 i+12 j

]=[

− τ2 − 1

2τ i + 12 j,

12τ + τ

2 i + 12 j]. Straightforward calculations show that

this extra element fixes the harmonic polynomial (3.32) if and only if

250 m. ionel

s = 2√

5r. Therefore, the harmonic cubics stabilized by I+ lie on theSO(4)-orbit of the polynomial r[x1(x2

1 − x22 − x2

3 − x24) + 2

√5x2x3x4].

To conclude, one can easily compute that the identity component ofthe stabilizer of the polynomial (3.32) is always discrete, except in thecase s = 0. q.e.d.

We now consider those special Lagrangian submanifolds L ⊂ C4

whose cubic fundamental form has a polyhedral symmetry at each point.

Theorem 3.12. Suppose that L ⊂ C4 is a connected special La-grangian 4-fold with the property that its fundamental cubic at each pointhas a tetrahedral symmetry T. Then, up to congruence and scaling, Lis the Harvey-Lawson example L ⊂ C4 defined in standard coordinatesby the equations

L : |z0| = |z1| = |z2| = |z3|Re(z0z1z2z3) = 5

√2.

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. Proposi-tion 3.11 implies that there exist functions r, s : L → R+ with s =

√5r

and a T-subbundle F ⊂ PL over L for which the following identity holds:

C = 3rω1(ω21 − ω2

2 − ω23 − ω2

4−) + 6sω2ω3ω4

Since F is an T-bundle, the following relations hold: α21 = α31 =α41 = α32 = α42 = α43 = 0 mod ω1, ω2, ω3, ω4. The usual differentialanalysis yields the following structure equations on F :

dω1 = 0(3.33)

dω2 =√

s2 − r2ω1 ∧ ω2

dω3 =√

s2 − r2ω1 ∧ ω3

dω4 =√

s2 − r2ω1 ∧ ω4

dr = −5r√

s2 − r2ω1

ds = −s√

s2 − r2ω1.

From the last two equations in (3.33), it follows that r = cs5, c > 0constant. We can suppose that c = 1 since the equations are invariantunder scaling. Moreover, s ∈ [−1, 1].


The above structure equations imply that ω1 = 0 defines an inte-grable 3-plane field which we denote by Γ2 and that ω2 = ω3 = ω4 = 0defines an integrable 1-plane field denoted by Γ1. Since dω1 = 0, itfollows that ω1 = dx1 on the leaves of the foliation Γ1. The structureequations (3.33) also imply d(sω2) = d(sω3) = d(sω4) = 0 and thereforethere exist functions x2, x3, x4 on L such that ω2 = dx2

s , ω3 = dx3s and

ω4 = dx4s . The metric g = dx2

2+dx23+dx2

4s2 is well-defined on the leaves of

the Γ2 foliation.Equations dei = ejαji − Jejβji and d(Jei) = ejβji + Jejαji yield

that, as matrices

d(e1 Je1) = (e1 Je1)(

0 3s5ω1

−3s5ω1 0

)mod ω2, ω3, ω4.

Therefore, the leaves of the Γ1 foliation are plane curves with curvaturek = 3s5, lying in the complex line (e1, Je1). These curves are congruentsince ds is a multiple of ω1.

Now consider the Γ2 foliation, defined by the equation ω1 = 0. Since

s is constant on its leaves, the connection matrix A =(

αij βij

−βij αij

)satisfies A ∧ A = dA = 0. Therefore A takes values in a 3-dimensionalabelian subalgebra g ⊂ su(4). The maximal torus of SU(4) is conjugateto the subgroup

T 3 =

diag(eiθ0 , eiθ1 , eiθ2 , eiθ3) |

3∑k=0

θi = 0 mod 2π

and the maximal torus acts on L by rotating around a plane curve C.Therefore, the solution is invariant under the torus action and the onlyspecial Lagrangian 4-folds with this property are described explicitly byHarvey and Lawson in their paper [12]. If (z0, z1, z2, z3) are coordinateson C4, then the special Lagrangian 4-folds in C4 invariant under T 3 looklike:

|z0|2 − |z1|2 = c1, |z0|2 − |z2|2 = c2, |z0|2 − |z3|2 = c3,(3.34)Re(z0z1z2z3) = a

for some real constants a, c1, c2, c3. It is easy to see that the solutionof the structure equations (3.33) is symmetric in (z1, z2, z3), thereforec1 = c2 = c3 = c.

252 m. ionel

Reparametrizing the solution using polar coordinates zk = rkeiθk ,

k = 0...3, (3.34) becomes

r20 − r2

1 = r20 − r2

2 = r20 − r2

3 = c(3.35)

θ0 = arccosa

r0r1r2r3− θ1 − θ2 − θ3.

We will find out for what constants a and c, the special Lagrangian4-fold defined by (3.35) is a solution of the structure equations. As wehave seen, the solution is a special Lagrangian 4-fold which is foliatedby congruent curves of curvature 3s5 and 3-manifolds which are theT 3-orbit of the points on the leaves of the first foliation.

Since z(r, θ1, θ2, θ3) = (√

c + r2, arccos ar3

√c+r2

− θ1 − θ2 − θ3, r, θ1, r,

θ2, r, θ3), the tangent plane to a T 3-orbits is spanned by the vectorsv1 = zθ1 = − ∂

∂θ0+ ∂

∂θ1, v2 = zθ2 = − ∂

∂θ0+ ∂

∂θ2and v3 = zθ3 = − ∂

∂θ0+ ∂

∂θ3.

We look now for another vector v0 in the tangent space of L suchthat v0, v1, v2, v3 are a basis of this tangent space. Since this tangentspace is special Lagrangian, the symplectic form ω and the imaginarypart of the holomorphic volume form Ω should vanish on it. Also, v0

should be orthogonal to vi, i = 1...3, so g(v0, vi) = 0 for i = 1...3.Let us write

v0 =3∑

i=0

µi∂

∂θi+ νi

∂

∂ri.

The symplectic form in polar coordinates is ω =∑3

i=0 ridri ∧ dθi andthe condition ω(v0, vi) = 0 for i = 1, 2, 3 implies that νi = r0ν0

ri= ν

ri,

where ν = r0ν0. The metric on C4 is g =∑3

i=0(dri)2 + r2i (dθi)2 and

the condition that g(v0, vi) = 0 for i = 1...3 implies the relations µi =µ0r2

0

r2i

= µr2i, for i = 1, 2, 3, where µ = µ0r

20. Finally, straightforward

calculations yield that the condition Im Ω(v1, v2, v3, v0) = 0 implies therelation ν

µ = tan(θ0 + θ1 + θ2 + θ3). Therefore,

v0 =3∑

i=0

1r2i

∂

∂θi+

tan(θ0 + θ1 + θ2 + θ3)ri

∂

∂ri.

Next, we find an integral curve of the vector field v0, that lies in:

L : r1 = r2 = r3 = r, r0 =√

c + r2,

θ0 = arccosa

r3√

c + r2− θ1 − θ2 − θ3.


When c = 0, an integral curve is given by

C : r0 = r1 = r2 = r3 = r, θ0 = θ1 = θ2 = θ3 = θ, cos(4θ) =a

r4.

Therefore, the curve C is given by: z1 = z2 = z3 = z4 = reiθ,r4 cos(4θ) = a and it is a plane curve which lies in the complex linez1 = z2 = z3 = z4. We have seen that L is foliated by plane curves withcurvature 3s5, so we will determine for what value of a the curve C hasthis curvature. We choose an orthonormal basis in the z1 = z2 = z3 = z4

plane: e1 = (12 , 0, 1

2 , 0, 12 , 0, 1

2 , 0) and e2 = (0, 12 , 0, 1

2 , 0, 12 , 0, 1

2) and inthis basis, the curve C is given by γ(θ) = (2r cos θ, 2r sin θ), where

r =(

acos(4θ)

) 14 .

Computing the curvature of γ, one gets k(θ) = −32a−

14 (cos(4θ))

54 .

But the curves that foliate L are parameterized by arclength

ω1 = dt = dθ|γ ′| = 2a14 (cos θ)−

54(3.36)

and the curvature in this parameterization is k = 3s5. Therefore

s = (k

3)

15 = − 1

215 a

120

(cos 4θ)14 .(3.37)

From the structure equations (3.33), it follows that ds = −s√

s2 − s10ω1

has to be satisfied. Using equations (3.36) and (3.37), we get

ds =1

265 a

310

(cos 4θ)12 sin 4θ ω1(3.38)

and from equation (3.38),

− s√

s2 − s10ω1(3.39)

=1

215 a

120

(cos 4θ)14

(1

225 a

110

(cos 4θ)12 − 1

4a12

(cos 4θ)52

) 12

ω1.

Equating these last two equations, it follows that a = 5√

2. The structureequations are satisfied now and L is a special Lagrangian 4-fold.

To conclude, the special Lagrangian submanifold L that is a solutionto the structure equations (3.33) can be described explicitly as:

L : |z0| = |z1| = |z2| = |z3|, Re(z0z1z2z3) = 5√

2.

q.e.d.

254 m. ionel

Theorem 3.13. Suppose that L ⊂ C4 is a connected special La-grangian 4-fold with the property that its fundamental cubic at eachpoint has an octahedral symmetry O+, where O+ is the irreducibly actingoctahedral subgroup of SO(4). Then, up to congruence, L is the Harvey-Lawson cone in C4 defined in standard coordinates (z0, z1, z2, z3) by theequation

L : |z0| = |z1| = |z2| = |z3|, Re(z0z1z2z3) = 0.(3.40)

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. Proposi-tion 3.11 implies that there exists a function s : L → R+ and an O+-subbundle F ⊂ PL over L for which the following identity holds:

C = 6sω2ω3ω4

and the 1-forms ω1, ω2, ω3, ω4 form a basis on F .Straightforward calculations show that the structure equations are:

dω1 = 0, dω2 = sω1 ∧ ω2, dω3 = sω1 ∧ ω3,(3.41)

dω4 = sω1 ∧ ω4, ds = −s2ω1.

The structure equations imply the equation

de1 = s(e2ω2 + e3ω3 + e4ω4) = s(dx − e1ω1),

where x : L+ → C4. From here and the last equation in (3.41), itfollows that x = e1

s + x0, where x0 is a constant which we can reduceto 0 by translation. Therefore x = e1

s . On the leaves of the foliationω2 = ω3 = ω4 = 0, de1 = 0 and thus the vector e1 is constant alongthese leaves. This tells us that the special Lagrangian 4-fold L+ is acone on some 3-dimensional manifold Σ ⊂ S7. We have to determinenow for what 3-dimensional manifolds Σ ⊂ S7, the cone C(Σ) is specialLagrangian and satisfies the structure equations.

In the case t = −s we obtain x = − e1s and the solution is again a

cone through the origin, call it L−. We have that L = L+ ∪ L−.

The connection matrix A =(

αij βij

−βij αij

)satisfies A ∧ A = dA = 0.

Therefore A takes values in an abelian subalgebra g ⊂ su(4). The groupG = exp g is a maximal torus of SU(4) and it is conjugate to the diagonaltorus T 3 =

diag(eiθ0 , eiθ1 , eiθ2 , eiθ3) :

∑3k=0 θi = 0 mod 2π

.


G acts transitively on the cone, so the cone is homogeneous and wehave to determine which of the orbits on the 7-sphere are special La-grangian. The solution is invariant under the torus action and thereforethe links of these special Lagrangian cones are 3-dimensional tori onS7. They are described explicitly by Harvey and Lawson in their paper[12]. It follows that L is given in standard coordinates (z0, z1, z2, z3) by(3.40).

Therefore, the special Lagrangian cone L is a union of two cones L+

(obtained in the case t = s) and L− (obtained in the case t = −s) withvertices at the origin through the 3-dimensional tori T+ and T− on S7

given by

T+ =(

12eiθ0 ,

12eiθ1 ,

12eiθ2 ,

12eiθ3

): θ0 + θ1 + θ2 + θ3 =

π

2

T− =

(12eiθ0 ,

12eiθ1 ,

12eiθ2 ,

12eiθ3

): θ0 + θ1 + θ2 + θ3 =

3π

2

.

Theorem 3.14. There are no connected special Lagrangian 4-foldswhose fundamental cubic at each point has an icosahedral symmetry I+,where I+ is the irreducibly acting icosahedral subgroup of SO(4).


C = 3r[ω1(ω21 − ω2

2 − ω23 − ω2

4) + 2√

5ω2ω3ω4]

defines an I+-subbundle F ⊂ PL of the L-adapted coframe bundle PL →L. The usual differential analysis on the subbundle F yields r = 0,contrary to the hypothesis. q.e.d.

3.3.2 Symmetries of order 6, 5 and 4

We have seen in Corollary 3.10 that the elements of a discrete stabilizerof a fundamental cubic of a special Lagrangian 4-fold have order less orequal to 6. From the proof of this corollary, the general harmonic cubicstabilized by an element of order 6 is

C = Re(rz31 + sz1z

22) = r(x3

1 − 3x1x22) + s[(x2

3 − x24)x1 − 2x2x3x4]

where we can arrange r, s to be real and nonnegative by making rotationsin the z1 and in the z2-lines. Easy computations show that the full

256 m. ionel

stabilizer of C is the dihedral group on 6 elements D6, if r = 0 ands = 0. A similar differential analysis as in the previous cases yields thefollowing result:

Theorem 3.15. There are no nontrivial special Lagrangian sub-manifolds in C4 whose fundamental cubic has a discrete stabilizer whichcontains an element of order 6.

Next, the general harmonic cubic stabilized by an element of order5 is

C = Re(rz1z22 + sz2

1z2)

where we can arrange r, s ≥ 0. The same kind of analysis gives:

Theorem 3.16. There are no nontrivial special Lagrangian sub-manifolds in C4 whose fundamental cubic has a discrete stabilizer whichcontains at least an element of order 5.

Remark. From this theorem, the result in Theorem 3.14 followsimmediately, since the irreducibly acting icosahedral subgroup of SO(4)has elements of order 5.

Next, the general harmonic cubic stabilized by an element of order4 is

C = Re(rz1z22 + sz1z

22)

where we can arrange again r, s to be real and nonnegative. The sta-bilizer of C is a continuous subgroup if r = 0 or s = 0, the irreduciblyacting octahedral subgroup O+ if r = s and the dihedral group D4 inthe rest of the cases, since the element of order 2 that flips the signs ofx2, x3 belongs to the stabilizer.

We obtain:

Theorem 3.17. There is no nontrivial special Lagrangian 4-fold inC4 whose fundamental cubic has a D4-symmetry at each point.

For the details of the calculations in the above results see [5].

3.3.3 Discrete symmetry at least Z3

Now we consider those special Lagrangian 4-folds L ⊂ C4 whose fun-damental cubic has at least a Z3-symmetry at each point. We saw inthe proof of Corollary 3.10 that there are two inequivalent orbits thatstabilize an element of order 3. We start with:


Case 1. (r, s) = (23 , 0) : The general harmonic cubic fixed by the

element g =(

e4πi3 00 I2

)in the maximal torus is:

C = Re(rz31 + tz3

2 + sz2(|z2|2 − 2|z1|2)),

where r, t, s ∈ C. By rotations in the z1-line and z2-line, we can arrangethat r, s be real and nonnegative. By writing t = u + iv, u, v ∈ R, thecubic C becomes:(∗)C = r(x3

1−3x1x22)+sx3(x3

3+x24−2x2

1−2x22)+u(x3

3−3x3x24)+v(x3

4−3x23x4)

where r, u, v, s ∈ R and r, s ≥ 0.The next lemma tells us what the full stabilizer of C is.

Lemma 3.18. The full stabilizer of the harmonic cubic polynomial(∗) is:

1) a continuous subgroup of SO(4), if r = 0 or s = u = v = 0;

2) the dihedral subgroup D3 generated by the order 3 element g andthe order 2 element that flips the signs of x2, x4, if r = 0, v = 0;

3) the dihedral subgroup D3 generated by the order 3 element g andthe order 2 element that flips the signs of x2, x4, if r = 0, v =0, u = 3s;

4) the order 18 normal subgroup of D3 × D3, if u = v = 0 andr, s = 0;

5) the cyclic subgroup Z3 generated by the order 3 element g if noneof the above relations among the parameters r, s, u, v hold.

Proof. We denoted by G be the stabilizer of the polynomial C,where r, s ≥ 0. A simple computation shows that G is a continuoussubgroup if and only if r = 0 or u = v = s = 0. Therefore, if r = 0 ands2 + u2 + v2 = 0, the stabilizer G is discrete.

When s = 0, we can make a rotation in the (x3, x4)-plane and sup-pose also that v = 0. In this case, the stabilizer of C is G, the order 18normal subgroup of D3 × D3 described as follows: Let the first D3 bedenoted by D3

+ and suppose it is generated by the rotation a1 and thereflection b1, where a3

1 = 1, b21 = 1, a1b1a1 = b1. Denote the second D3

258 m. ionel

by D3− and suppose it is generated by the rotation a2 and the reflec-

tion b2, where a32 = 1, b2

2 = 1, a2b2a2 = b2. Then D3+ consists of the

elements

θ+1 = 1, θ+

2 = a1, θ+3 = a2

1, r+1 = b1, r+

2 = a1b1, r+3 = a2

1b1

and D3− consists of the elements

θ−1 = 1, θ−2 = a2, θ−3 = a22, r−1 = b2, r−2 = a2b2, r−3 = a2

2b2.

The SO(4)-stabilizer of the cubic C is formed by the 18 pair elements:

(θ+i , θ−j ), (r+

i , r−j ), i, j = 1...3

Next, if r = 0 and s = 0, the differential analysis yields the followingcases:

i) If v = 0, the stabilizer G of the cubic C is the dihedral subgroupD3 generated by the order 3 element g and the order 2 elementthat flips the signs of x2 and x4.

ii) If v = 0, u = 3s, the stabilizer G of C is also the above dihedralsubgroup D3.

iii) In the general case, when none of the above relations among theparameters r, s, u, v hold, the stabilizer of C is Z3.

q.e.d.

In the case of D3-symmetry we obtain the following partial result:

Proposition 3.19. There is an infinite parameter family of con-nected special Lagrangian submanifolds in C4 such that the fundamentalcubic at each point has a D3-symmetry and is of the form (∗), wherev = 0 and r, s = 0. This family depends on 2 functions of one variable.

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. It followsthat

C = r(ω31 − 3ω1ω

22) + u(ω3

3 − 3ω3ω24) + 3sω3(ω2

3 + ω24 − 2ω2

1 − 2ω22),

with r > 0, s > 0 defines a D3-subbundle F ⊂ PL of the adaptedcoframe bundle PL → L. In this case, were able to write down thestructure equations that hold on the bundle F , but were unable todescribe completely the family of special Lagrangian submanifolds inthis case. Cartan-Kahler theorem tells us that the family should dependon 2 functions of one variables. For more details see [5]. q.e.d.


Theorem 3.20. Let L be a connected special Lagrangian subman-ifolds in C4 such that its fundamental cubic at each point has a D3-symmetry and it is of the form (∗), where v = 0, u = 3s and r, s = 0.Then L is, up to rigid motion, an open subset of the asymptoticallyconical special Lagrangian 4-fold given by:

LΣ = (a + ib)u| u ∈ Σ, Re(a + ib)4 = c,(3.42)

where c is a real constant and Σ ⊂ S7 is a 3-manifold with the propertythat the cone on it is special Lagrangian, with phase i.

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. It followsthat

C = r(ω31 − 3ω1ω

22) + 3sω3(ω2

3 − ω21 − ω2

2 − ω24),

with r > 0, s = 0 defines a D3-subbundle F ⊂ PL of the adaptedcoframe bundle PL → L.

The structure equations on this bundle are computed to be:

dω1 = t1ω3 ∧ ω1 − t2ω4 ∧ ω1 − 2t5ω4 ∧ ω2 + t3ω1 ∧ ω2

(3.43)

dω2 = t4ω1 ∧ ω2 − t1ω2 ∧ ω3 + t2ω2 ∧ ω4 + 2t5ω4 ∧ ω1

dω3 = 0dω4 = 6t5ω1 ∧ ω2 + t1ω3 ∧ ω4

dr = −3rt4ω1 + 3rt3ω2 − rt1ω3 + rt2ω4

ds = −5st1ω3

dt1 = (4s2 − t21)ω3

dt2 = m1ω1 + m2ω2 − t1t2ω3 + (t21 + t22 + s2 − 9t25)ω4

dt3 = m3ω1 + (m4 − 2r2 + t21 + t22 + t23 + t24 + 15t25 + s2)ω2 − t1t3ω3

+(

t2t3 − 2t4t5 +13m2

)ω4

dt4 = m4ω1 − (m3 + 2t2t5)ω2 − t1t4ω3 +(

2t3t5 + t2t4 −13m1

)ω4

dt5 =13m2ω1 −

13m1ω2 − t1t5ω3 + 2t2t5ω4

for some functions m1, m2, m3, m4. Differentiation of these equationsdoes not lead to new relations among the quantities. The differential

260 m. ionel

ideal on the manifold M = PL × R3 is involutive, since the Cartancharacters can be computed as s1 = 4, s2 = s3 = s4 = 0 and the spaceof integral elements at each point is parameterized by the 4 parametersm1, m2, m3, m4.

The structure equations imply d(s85 + t21s

− 25 ) = 0. Since F and

L are connected, it follows that there exists a constant c > 0 so thats

85 + t21s

− 25 = c

85 . Therefore, there is a function θ, well-defined on L,

that satisfies:

s45 = c

45 cos 4θ, s−

15 t1 = c

45 sin 4θ, |θ| <

π

8.

From the sixth equation of (3.43), it follows that

ω3 =dθ

c(cos 4θ)54

.

The structure equations imply that ω1 = ω2 = ω4 = 0 is integrableand also that ω3 = 0 defines an integrable 3-plane field on L. The1-dimensional leaves of the field Γ1 defined by ω1 = ω2 = ω4 = 0 arecongruent along Γ2, the codimension 1 foliation defined by ω3 = 0. Thisis clear since:

de3 = −3sω3Je3, d(Je3) = 3sω3e3 mod ω1, ω2, ω4

and ds = 0 mod ω3, meaning s is constant along each leaf of Γ2. Theabove equations imply that the leaves of the Γ1 foliation are congruentplane curves of curvature −3s, lying in the complex line (e3, Je3).

The form of the structure equations tells us that these examplesmust be related to the asymptotically conical special Lagrangian sub-manifolds, as seen in [1] for the Z3-stabilizer type case of the specialLagrangian 3-folds.

Suppose that the plane curves which are the leaves of the Γ1 foliationare of the form Re z

1p = c

1p , where c is a constant and p ∈ R is to be

determined. By dilation, we can take c = 1 and consider the curve givenby z(t) = (1 + it)p, in the (e3, Je3)-plane. To compute the curvature ofthis curve, we use the formula for the curvature in any parametrizationand we get:

k(t) =z′ ∧ z′′

(dsdt )

3e3 ∧ Je3

=p − 1

p(1 + t2)−

p+12 .(3.44)


Since k = −3s, and also using the sixth and the seventh structureequations in (3.43), we compute that p = 1

4 .Therefore, the leaves of the Γ1-foliation are curves given by the equa-

tion Re z4 = c, where c is a constant. From the equation of the curva-ture (3.44), it follows that as t → ∞, k → 0, so these curves flatten out,telling us that they have an asymptote.

Now we study the Γ2-foliation, whose leaves are 3-manifolds. If weset θ = 0, i.e., t1 = 0 and s = c, we obtain a 3-manifold Σ, immersed inthe 7-sphere S7. This is clear since:

d(Je3) = −sω1e1 − sω2e2 − sω4e4 = −sdx,

where x : Σ → C4 is the position vector. Since s is constant on Σ, itimplies that

Je3 = −sx + constant,

where we can suppose, by translation, that the constant is 0. Therefore,x = −Je3

s and Σ is immersed in the 7-sphere of radius 1s , in the direction

Je3.The structure equations of the leaves of the ω3 = 0 foliation are:

dω1 = −t2ω4 ∧ ω1 − 2t5ω4 ∧ ω2 + t3ω1 ∧ ω2

dω2 = t4ω1 ∧ ω2 + t2ω2 ∧ ω4 + 2t5ω4 ∧ ω1 mod ω3

dω4 = 6t5ω1 ∧ ω2.

Consider now the following expressions:

ηi = s15 ω1, i = 1, 2, 4,

qi = s−15 ti, i = 2, . . . , 5,

p = s−15 r,

vi = s−25 mi, i = 1, . . . , 4.

The structure equations derived earlier show that

dη1 = q2η1 ∧ η4 + q3η1 ∧ η2 + 2q5η2 ∧ η4(3.45)dη2 = q2η2 ∧ η4 + q4η1 ∧ η2 + 2q5η4 ∧ η1

dη4 = 6q5η1 ∧ η2

dq2 = v1η1 + v2η2 +(

49

+ q22 − 9q2

5

)η4

262 m. ionel

dq3 = v3η1 +(

49

+ q22 + q2

3 + q24 + 15q2

5 − 2p2 + v4

)η2

+13(v2 − 6q4q5 + 3q2q3)η4

dq4 = v4η1 − (v3 + 2q2q5)η2 −13(v1 − 6q3q5 − 3q2q4)η4

dq5 =13v2η1 −

13v1η2 + 2q2q5η4

dp = −p(3q4η1 − 3q3η2 − q2η4).

Therefore, the metric g = η21 +η2

2 +η24 is well-defined on each leaf of the

Γ2-foliation. The θ-curves meet the 3-manifold Σ orthogonally, so it iseasy to see that the image of (−π

8 , π8 ) × Σ is of the form:

LΣ = zu | u ∈ Σ, z ∈ C, Re z4 = c,(3.46)

where c is a real constant. In order for this to be a special Lagrangian4-fold, the cone on the image of Σ should be a special Lagrangian 4-fold.

We shall show now that, indeed, the cone on Σ is special Lagrangianwith phase i. The cone on Σ is parameterized by:

(r, z) → rz, r ∈ R+, z ∈ Σ3.

The tangent space to C(Σ) has a basis formed by the vectors:(e1 =

∂

∂x1, e2 =

∂

∂x2, e4 =

∂

∂x4, Je3 =

∂

∂y3

).

Sinceω = dx1 ∧ dy1 + dx2 ∧ dy2 + dx3 ∧ dy3 + dx4 ∧ dy4,

it is clear that ω |C(Σ)= 0, so the cone is Lagrangian. Also, Ω =dz1 ∧ dz2 ∧ dz3 ∧ dz4 and we can easily compute that

Im Ω |C(Σ)= dx1 ∧ dx2 ∧ dy3 ∧ dx4,

which represents the volume form on the cone, and Re Ω |C(Σ)= 0.Therefore, C(Σ) is special Lagrangian with phase i. Then, it is well-known [12] that (3.46) is a special Lagrangian 4-fold.

Theorem 3.21. Let L be a connected special Lagrangian subman-ifolds in C4 such that its fundamental cubic at each point has a G-symmetry, where G is the order 18 normal subgroup of D3 ×D3. ThenL is congruent to the product of two holomorphic curves in C2.


Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. Then

C = r(ω31 − 3ω1ω

22) + v(ω3

3 − 3ω3ω24),

with r > 0, v > 0, r = v defines a G-subbundle F ⊂ PL of the adaptedcoframe bundle PL → L, where G is the order 18 normal subgroup ofD3 × D3.

The structure equations on the subbundle F are computed to be:

dω1 = t3ω1 ∧ ω2, dω2 = t4ω1 ∧ ω2(3.47)dω3 = t1ω3 ∧ ω4, dω4 = t2ω3 ∧ ω4

dr = −3rt4ω1 + 3rt3ω2

dv = −3vt2ω3 + 3vt1ω4

dt1 = u1ω3 + (t21 + t22 − 2v2 + u2)ω4

dt2 = u2ω3 − u1ω4

dt3 = u3ω1 + (t23 + t24 − 2r2 + u4)ω2

dt4 = u4ω1 − u3ω2

for some functions u1, u2, u3, u4.From the above structure equations, we can see that ω1 = ω2 = 0

and ω3 = ω4 = 0 define integrable 2-plane fields on L. The structureequations also show that the leaves of the 2-plane field Γ1 defined byω3 = ω4 = 0 are congruent along Γ2, the codimension 2 foliation definedby ω1 = ω2 = 0. Also, the 2-dimensional leaves of the 2-plane field Γ2

are congruent along Γ1.Since d(e1ω1 + e2ω2) = 0 and d(e3ω3 + e4ω4) = 0, it follows that

e1ω1+e2ω2 = dπ1 and e3ω3+e4ω4 = dπ2, where the projections π1 : L →Σ1 and π2 : L → Σ2 are well-defined. Therefore, x = π1 + π2 + constand L is the sum of two surfaces: L = Σ1 × Σ2.

Since d(e1 ∧ e2 ∧ Je1 ∧ Je2) = 0, it follows that Σ1 lies in thecomplex plane (e1, e2, Je1, Je2). Also, Σ2 lies in the complex plane(e3, e4, Je3, Je4).

Because L is special Lagrangian, both surfaces Σ1 and Σ2 shouldbe special Lagrangian 2-folds in C2. It is well-known then that thesesurfaces should be holomorphic curves with respect to some complexstructure on C2. More explicitly, if Σ1 ⊂ C2, with complex coordinatesz1 = x1 + iy1, z2 = x2 + iy2, then Σ1 is a holomorphic curve withrespect to the complex coordinates u1 = x1 − ix2, v1 = y1 + iy2.

264 m. ionel

If Σ2 ⊂ C2, with standard complex coordinates z3 = x3 + iy3, z4 =x4 + iy4, then Σ2 is a holomorphic curve with respect to the complexcoordinates u2 = x3 − ix4, v2 = y3 + iy4. q.e.d.

In the next case of Z3-symmetry we were unable to describe com-pletely the SL 4-folds and therefore we have only a partial result.

Proposition 3.22. There is an infinite parameter family of con-nected special Lagrangian submanifolds in C4 such that the fundamentalcubic at each point has a Z3-symmetry and is of the form (∗). Thefamily depends on 4 functions of one variable and the elements of thisfamily are foliated by non-congruent minimal Legendrian surfaces in thedirection ω1, ω2 and by congruent holomorphic curves in the directionω3, ω4.

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. Then

C = r(ω31 − 3ω1ω

22) + 3s(ω2

3 + ω24 − 2ω2

1 − 2ω22)ω3

+ u(ω33 − 3ω3ω

24) + v(ω3

4 − 3ω23ω4)

with r > 0 defines a Z3-subbundle F ⊂ PL of the adapted coframebundle PL → L. The differential analysis shows that the structureequations on the bundle F are:

dω1 = t3ω1 ∧ ω2, dω2 = t4ω1 ∧ ω2, dω3 = t1ω3 ∧ ω4, dω4 = t2ω3 ∧ ω4,

(3.48)

dr = −3rt4ω1 + 3rt3ω2 + 2rst2ω3 + 2rst1ω4

ds = [−svt1 + s(7s + u)t2]ω3 + s[(3s − u)t1 − vt2]ω4

du = [−11svt1 + (3v2 + 3u2 + 6s2 − su)t2 + t6]ω3

+ [(11su − 3u2 − 6s2 − 3v2)t1 − svt2 − t5]ω4

dv = t5ω3 + t6ω4

dt1 = [v(t21 + t22 + 1) − 8st1t2]ω3 + [(u − s)(t21 + t22 + 1)]ω4

dt2 = −[(u + 5s)(t21 + t22 + 1) − 8st21]ω3 + v(t21 + t22 + 1)ω4

dt3 = −m2ω1 + [4s2(t21 + t22 + 1) + t23 + t24 − 2r2 + m1)ω2

+ 2st2t3ω3 + 2st1t3ω4

dt4 = m1ω1 + m2ω2 + 2st2t4ω3 + 2st1t4ω4


dt5 =[− m4 + 3t21(30suv − 47s2v − 3u2v − 3v3)

+ 3t22(−7s2v + 10suv − 3v3 − 3u2v)

+ (25s − 7u)t1t6 + 60sv2t1t2 + (7u − 3s)t2t5− 7vt2t6 − 7vt1t5 − 18s2v − 6u2v + 24suv − 6v3

]ω3

+ [m3 + (3s − u)t1t5 − vt2t5 + (s − u)t2t6 + vt1t6]ω4

dt6 = m3ω3 + m4ω4

for some functions m1, m2, m3, m4.The Cartan-Kahler analysis tells us that the solution should depend

on 4 functions of 1 variable. From the structure equations, we can seethat ω3 = ω4 = 0 and ω1 = ω2 = 0 define integrable 2-plane fields onL. Let Γ1 be the ω1 = ω2 = 0 foliation and Γ2 be the ω3 = ω4 = 0foliation. The structure equations of the foliation Γ1 show that theleaves are congruent and that the metric g1 = ω2

3 + ω24 is well-defined

on the leaf space of the Γ1 foliation. It is easy to see that the leaves ofthe Γ2 foliation are non-congruent.

Notice that if we denote ∆2 = 4s2(t21 + t22 + 1), then we get thatd∆∆ = 2s(t2ω3 + t1ω4). We see that ∆ is constant on each leaf of the Γ2

foliation. We compute that:

d(∆ω1) = t3ω1 ∧ (∆ω2) = (t3ω1 + t4ω2) ∧ (∆ω2)(3.49)d(∆ω2) = −t4ω2 ∧ (∆ω1) = −(t3ω1 + t4ω2) ∧ (∆ω1)

and the metric g2 = (∆ω1)2 + (∆ω2)2 is well-defined on the leaf spaceof the Γ2 foliation.

Computations also show that d(r13 ∆

23 ω1) = 0 and d(r

13 ∆

23 ω2) = 0.

These imply that there are functions x1, x2 on L such that r13 ∆

23 ω1 =

dx1 and r13 ∆

23 ω2 = dx2. This gives x1, x2 up to additive constants

and x1 + ix2 is holomorphic with respect to the complex structure thatω1, ω2 define on the leaf space of Γ2.

The above imply that the metric

g2 =(

∆r

) 23

(dx21 + dx2

2) = F (x1, x2)(dx21 + dx2

2),

on the Γ2 leaf space has Gauss curvature:

k = 1 − 2( r

∆

)2= 1 − 2

F 3(x1, x2).

266 m. ionel

We obtain a differential equation for the function F (x1, x2), given by:

12∆(lnF ) =

2F 2

− F.(3.50)

We used here the fact that a metric ds2 = e2u(dx2+dy2) is computed tohave the Gauss curvature K = −∆ue−2u. In our case, take u = 1

2 lnFand (3.50) follows. The function u satisfies the differential equation∆u = e2u − 2e−4u (Tzitzeica equation), which is completely integrableby means of inverse scattering method [16]. This is the differentialequation satisfied by the curvature of the metric of a minimal Legendrianimmersion in S5(1), invariant under S1-action, as shown by Mark Haskinin [4], p. 14. Sharipov [16] shows that the minimal immersion satisfyingTzitzeica equation are minimal tori which are complexly normal in S5.Therefore, L is foliated by non-congruent minimal Legendrian surfacesin the direction ω1, ω2 and by congruent holomorphic curves in thedirection ω3, ω4. We do not have a complete description of the familyyet.

We move now to analyze the other orbit that stabilizes an elementof order 3.

Case 2. (r, s) = (23 , 1

3): This case is equivalent to the (r, s) = (13 , 1

3)case, when the element in the maximal torus that stabilizes C is

g =(

e2πi3 0

0 e2πi3

),

of order 3. The general harmonic cubic fixed by this element is

C = Re(a3z31 + 3a2z

21z2 + 3a1z1z

22 + a0z

32), a0, a1, a2, a3 ∈ C.

We notice that the commutator of g is larger than the maximal torus inthis case. The unitary group U(2) commutes with g and therefore wecan use also its action to get rid of certain parameters, more preciselyto make a0 = 0 and a1, a2 ∈ R. So, the general cubic stabilized by gwill look like:

(∗∗) C = u(x31 − 3x1x

22) + v(3x2

1x2 − x32)

+ r[(x21 − x2

2)x3 − 2x1x2x4] + s[(x23 − x2

4)x1 − 2x2x3x4],

where u, v, r, s ∈ R.


Lemma 3.23. The full stabilizer of the polynomial given by (∗∗),where r, s, u, v ∈ R is:

1) a continuous subgroup of SO(4), if r = s = 0 or u = v = r = 0 oru = v = s = 0;

2) the dihedral subgroup D6 generated by the order 6 element a =(e

4πi3 0

0 eπi3

)and the element of order 2 that flips the signs of x2,

x4, if r = v = 0;

3) the dihedral subgroup D3 generated by the order 3 element g andthe order 2 element that flips the signs of x2, x4, if s = v = 0;

4) the cyclic subgroup Z3 generated by the order 3 element g if noneof the above relations among the parameters r, s, u, v hold.

Proof. We denoted by G be the stabilizer of the polynomial C. Asimple computation shows that G is a continuous subgroup if and onlyif r = s = 0 or u = v = r = 0 or u = v = s = 0.

Doing the differential analysis in the discrete case, we obtain thefollowing cases where the stabilizer becomes larger than Z3:

i) r=0. By making a rotation, if necessary, of angle θ= 13 arctan(− v

u)in the (x1, x2)-plane and of angle −2θ in the (x3, x4)-plane, we cansuppose that v = 0 also.

The stabilizer of C for r = v = 0 is seen to be the dihedral subgroupD6 generated by the order 6 element a =

(e

4πi3 0

0 eπi3

)and the element of

order 2 that flips the signs of x2, x4. This case of symmetry at leastZ6 was already studied in Section 3.3.2 and it did not yield any familiesof special Lagrangian 4-folds.

ii) s = 0. In this case, we can arrange that v = 0 also and thestabilizer is computed to be the dihedral group D3.

iii) In the general case, when none of the above relations amongthe parameters r, s, u, v hold, the stabilizer of C is computed to be Z3

generated by the element g. q.e.d.

Theorem 3.24. Let L be a connected special Lagrangian subman-ifold in C4 such that its fundamental cubic at each point has a Z3-symmetry and it is of the form (∗∗). Then L is an I-special LagrangianJ-holomorphic surface in C4, where I, J, K is the hyper-Kahler stuc-ture on C4.

268 m. ionel

Proof. Let L be a special Lagrangian 4-fold that satisfies the hy-potheses of the theorem and let C be its fundamental cubic. FromLemma 3.23, the equation

C = u(ω31 − 3ω1ω

22) + v(3ω2

1ω2 − ω32)

+ r[(ω21 − ω2

2)ω3 − 2ω1ω2ω4]) + s[(ω23 − ω2

4)ω1 − 2ω2ω3ω4],

with r, s, u, v ∈ R defines a Z3-subbundle F ⊂ PL of the adapted coframebundle PL → L.

On the subbundle F , the following identities hold:

(βij)

(3.51)

=

uω1 + vω2 + rω3 vω1 − uω2 − rω4 rω1 + sω3 −rω2 − sω4

vω1 − uω2 − rω4 −uω1 − vω2 − rω3 −rω2 − sω4 −rω1 − sω3

rω1 + sω3 −rω2 − sω4 sω1 −sω2

−rω2 − sω4 −rω1 − sω3 −sω2 −sω1

.

The Cartan-Kahler analysis yields the following relations betweenthe αij ’s:

α31 − α42 = 0 and α32 + α41 = 0.

We consider the ideal I1, on the coframe bundle, spanned by the1-forms (3.51) and the two 1-forms α31 − α42 and α32 + a41. The inde-pendence condition is given by ω1 ∧ ω2 ∧ ω3 ∧ ω4 = 0 and the tableaumatrix for the structure equations is given by:

α1 α2 α3 α4

α2 −α1 α4 −α3

α3 α4 α5 α6

α4 −α3 α6 −α5

α5 α6 −3sπ6 −3sπ5

α6 −α5 −3sπ5 3sπ6

where

π1 = dr, π2 = ds, π3 = du, π4 = dv,

π5 = α41, π6 = α42, π7 = α43, π8 = α21

α1 = −π3 + 3rπ6 + 3vπ8


α2 = −π4 − 3rπ5 − 3uπ8

α3 = −π1 + vπ5 + (2s − u)π6

α4 = −(2s + u)π5 − vπ6 − rπ7 − 2rπ8

α5 = −2rπ2 − π6

α6 = −2rπ5 − 2sπ7 + sπ8.

From the above tableau, we compute the reduced Cartan characters ass′1 = 6, s′2 = 2, s′3 = s′4 = 0. The integral elements of the system at eachpoint is shown to form a space of dimension 10 = s′1 + 2s′2 + 3s′3 + 4s′4and therefore, by Cartan’s Test, the system I1 is involutive.

The form of the tableau resembles the tableau for the structureequations of a complex surface with complex structure given by γ1 =ω1 + iω2 and γ2 = ω3 + iω4. We will show that this is actually thecase. The characteristic variety of the ideal is formed by 2 complexlines spanned by γ1, γ2 and their conjugates.

The first derived system of I1 is generated by the rank 6 Pfaffiansystem I2 spanned by the six 1-forms:

θ1 = β11 + β22, θ2 = β33 + β44, θ3 = β41 − β32(3.52)θ4 = β31 + β42, θ5 = α31 − α42, θ6 = α32 + α41.

This system is Frobenius and defines a foliation of dimension 10 on thecoframe bundle. The integral manifold of our original system will be asubmanifold of the maximal integral manifold of the derived system I2.Therefore, we will adapt frames and restrict to the first derived system,looking for integral manifolds of this system.

We notice that, when restricted to the first derived system, the con-nection matrix takes values in the Lie algebra of a 10-dimensional sub-group of SU(4). This subgroup can be shown to be Sp(2). The systemI2 restricts to the Sp(2)-coframe bundle, of dimension 18. The canonicalform on this bundle has components ξi = ωi + iηi and the 1-forms

ωi, ηi, β11, β33, β21, β31, β41, β43, α21, α31, α41, α43

form a basis for the space of 1-forms on this coframe bundle P ∼= C4 ×Sp(2). On the integral manifolds, ηi = 0 for i = 1...4.

Now, the symplectic group Sp(2) leaves invariant 3 symplectic 2-forms ζ1, ζ2, ζ3. One of them is the Kahler form of the standard

270 m. ionel

complex structure I on R8.

ζ1 =i

2(ξ1 ∧ ξ1 + ξ2 ∧ ξ2 + ξ3 ∧ ξ3 + ξ4 ∧ ξ4)

= ω1 ∧ η1 + ω2 ∧ η2 + ω3 ∧ η3 + ω4 ∧ η4

and the other 2-forms are computed to be:

ζ2 = ω1 ∧ ω2 + ω3 ∧ ω4 − η1 ∧ η2 − η3 ∧ η4

ζ3 = ω1 ∧ η2 − ω2 ∧ η1 + ω3 ∧ η4 − ω4 ∧ η3.

Let I, J, K be the complex structures on R8 corresponding to leftmultiplication by the elementary quaternions i, j and k. Then thestandard metric g =

∑4i=1(ω

2i + η2

i ) on R8 is Kahler with respect toeach I, J, K, with Kahler form ζ1, ζ2 and ζ3, respectively. The formsψ1 = ζ2 + iζ3, ψ2 = ζ1 + iζ3, ψ3 = ζ1 + iζ2 are the holomorphic sym-plectic forms on C4, associated to the complex structures I, J and Krespectively. The standard complex structure on R8 is considered tobe I, given by the complex 1-forms: ωj + iηj , j = 1...4. The 4-formsΩi = 1

2ψ2i , i = 1...3 are the holomorphic volume forms on C4, associ-

ated to the complex structures I, J and K, respectively, with Ω1 beingthe usual holomorphic volume form. On the integral manifolds of I2,ζ1 = ζ3 = 0 and ζ2 = ω1 ∧ ω2 + ω3 ∧ ω4 is the Kahler form for thecomplex structure J given by γ1 = ω1 + iω2 and γ2 = ω3 + iω4.

We will now show that the integral manifold of the ideal generatedby the 2-forms ζ1 and ζ3 are complex manifolds with respect to thecomplex structure J . Let (z1, z2, z3, z4) be the complex coordinates onR8 that are holomorphic for the complex structure J . Then:

ζ1 + iζ3 = dz1 ∧ dz2 + dz3 ∧ dz4

Let I be the differential ideal generated by the complex 1-form ψ2 =ζ1 + iζ3. We use the Cartan-Kahler analysis [1] to compute the Cartancharacters as s1 = s2 = 2 and s3 = s4 = 0. The space of 2-dimensionalintegral elements over a point has dimension 6 = s1 + 2s2 + 3s3 + 4s4

and by Cartan’s Test, the system is involutive. The maximal integralmanifolds of this ideal are given by 2 complex linear equations, i.e., theyare J-holomorphic surfaces in C4.

An integral manifold of the derived system I2 is an integral manifoldof the system ζ1 = ζ3 = 0 and an integral manifold of the system ζ1 =ζ3 = 0 is an integral manifold of the derived system. To summarize, the


integral manifolds Σ of our original system are J-holomorphic surfacesin C4. They are I-special Lagrangian 4-folds, because

ζ1 |Σ= 0 and12

Im(Ω21) |Σ= ζ2 ∧ ζ3 |Σ= 0.

q.e.d.

Theorem 3.25. Let L be a connected special Lagrangian subman-ifold in C4 such that its fundamental cubic at each point has a D3-symmetry and it is of the form (∗) with s = v = 0. Then L is a ruledI-special Lagrangian J-holomorphic surface in C4.

Proof. The analysis here is similar to the one in the previous result.It can be shown that the solutions are again I-special Lagrangian J-holomorphic surfaces. Moreover, the structure equations show that theholomorphic surfaces are foliated by planes in the e3, e4-direction.The conclusion is that the solutions are ruled I-special Lagrangian J-holomorphic surfaces. q.e.d.

We conclude this paper with the following:

Open Problem. It remains to study the general case when thesymmetry of the fundamental cubic is at least a Z2. This is the mostcomplicated case since the space of fixed harmonic cubics involves alarge number of parameters.

References

[1] R.L. Bryant, Second order families of special Lagrangian 3-folds, math.DG/0007128.

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McMaster University






http://nyjm.albany.edu:8000/jdg/2001/58-1-1.htm

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